When we're doing work in apologetics, evangelism, politics, or some other context, it's important to be well informed about the state of the culture we're interacting with. I want to recommend some resources. I can't be exhaustive, and I won't be saying much about these sources. But you can search the Triablogue archives or my posts on Facebook, for example, to find further discussion of the significance of these sources and their findings. I won't always link the latest research. Some of these sources are ones I don't consult every year, and there's only so much that I've read from these sites.
The Pew Research Center publishes a lot of material on relevant subjects. Go here for an article on where Americans find meaning in life, for example. Among other results, they reported, "Overall, 20% of Americans say religion is the most meaningful aspect of their lives, second only to the share who say this about family (40%)."
Around this time every year, the Department of Labor publishes their annual research on how Americans spend their time. See here. I've been following their research for several years, and they've consistently found that the average American spends more than five hours a day on what they call leisure and sports and less than ten minutes a day on religious and spiritual activities. Here's one of many posts over the years in which I've discussed the implications of those findings.
Gallup has a lot of useful information. Here's a collection of resources on moral issues. The page here shows you how Americans' views on moral issues have changed over time. And here's an article that discusses why acceptance of polygamy has been growing.
There's been a major reduction in global poverty in recent decades. See here and here. That has major implications for how concerned we should be about poverty, how much attention we should give it, how much Christians should be focused on poverty in the local church and other contexts (e.g., financial giving), the proper size and role of government programs addressing poverty, etc. In my experience, the vast majority of people seem unaware of statistics like these or haven't thought much about the implications of them.
The Annenberg Public Policy Center does a lot of research on how much Americans know about civics issues. See here, for example. C-SPAN has commissioned polling related to the Supreme Court. Their 2018 poll was done at the time of Brett Kavanaugh's nomination to the Supreme Court. A majority of Americans couldn't name a single Supreme Court justice.
Some of these sites have a lot of data on religious issues. See, for example, here on Bible reading, here on how Americans view the Bible, and here regarding their views on issues related to Christmas. On global percentages for religious affiliation, see here. One important fact to note from the page I just cited and others at these web sites is that atheists make up such a small percentage of the population. We should keep that in mind when considering issues like how much of our apologetic effort should be directed toward atheism. Barna does a lot of research on Evangelicals in particular. Another site has an article on church attendance numbers.
Showing posts with label Statistics. Show all posts
Showing posts with label Statistics. Show all posts
Monday, June 22, 2020
Wednesday, April 01, 2020
Tuesday, March 24, 2020
Spreading fear, intentionally or unintentionally
 |
| Is there really a plague upon us all? |
From the look of it, it appears as if there is “a plague upon us all”. Certainly uncontrolled outbreaks of novel (new) viruses can get that way, and they have, historically, in real life, been incredibly damaging, and incredibly frightening.
But some charts and maps (like this one) aren’t helping us really to understand what’s going on, and where we are in real life.I think that the way the graphics are set up, they are more inclined to spread fear, than to explain things accurately.
It’s true that we don’t yet have all the best numbers, because not everyone’s testing is either accurate or adequate. I trust that’s a situation that is correcting itself as things move forward. In some cases, such as South Korea, we do have that kind of numbers. In other places, maybe not.
But as better information becomes distributed, we can expect that we’ll be able to see more clearly those locations where the curve is flattening, and where it is not, and the things they are doing to help things along. The video here, though it’s a few weeks old, will give you an idea of what to look for.
Exponential charts can’t work in real life.
Here’s a video that shows why that’s so, and when and how to expect and understand when things begin “flattening”.
How Exponential Growth Charts Work
Thursday, December 15, 2016
Saturday, November 01, 2014
"Polling Data Consistently Shows Women are Pro-Life on Abortion"
"Ironically, it turns out that women are much more supportive of the fictitious 'war on women' than men."
Read the rest here.
Tuesday, September 02, 2014
Breaking Littlewood's Law
Some atheists invoke "Littlewood's Law" to dismiss miracles as statistically inevitable cases of sheer coincidence. There are books on the subject which popularize that outlook.
Problem is, facile appeal to"Littlewood's Law" proves too much. They render cheating undetectable. Sometimes the dice are loaded. Sometimes the deck is stacked:
Tuesday, February 18, 2014
Saturday, June 22, 2013
How Americans Use Their Time
I want to pass along something I heard about on Michael Medved's radio program yesterday. The Department of Labor recently released a study of Americans' time management. See here regarding Americans' leisure time, for example.
Friday, September 23, 2011
Rabble Rauser
FWIW, if anything, I left the following three comments in the combox of Randal Rauser's post "Patrick Chan of Triablogue defends Randal Rauser with biting satire." But at least at the time of this post, the third comment doesn't seem to have gone through.
Wednesday, April 27, 2011
Undesigned coincidences
Tim McGrew writes the following:
HT: Steve.
Folks,BTW, the MP3 is available for download here.
Some of you have expressed an interest in hearing my Easter Sunday radio interview on undesigned coincidences as evidence for the reliability of the Gospels. That interview is now online. Go to
http://evidence4faith.com/
and click on "Podcasts" -- it should start playing automatically, and this week it's at the top of the queue. Later it will be pushed down in the list (the most recent episode is always at the top), but it should be there long-term.
Enjoy!
Tim
HT: Steve.
Sunday, February 15, 2009
How Popular Is Atheism In The United States?
Here's a thread on the lack of growth of atheism in the United States.
Monday, May 12, 2008
Statistical Analysis
Even if you only casually read through news websites (such as those of CNN or FOXNews), several times per month you will notice headlines such as the following:
The existence of so many studies helps to emphasize a point regarding statistical analysis. Despite being a powerful tool, if you do not set up the guidelines and restrictions for your samples properly any statistics you observe won’t amount to a hill of beans. And we’re not even talking about the inherent fluctuations that require the existence of error bars (that’s the line that says +/- 3%, for example). Nor are we even addressing political manipulation of statistics in the form of pollaganda. Instead, I’m talking about something at the heart of statistics itself—it’s a universal.
To demonstrate what it is, let us first ask a simple question. When we do a statistical analysis of some observation, for what reason are we doing it? As you can see in the above headline examples, most of the time studies are done to find a causal linking between some object and/or action and some result. Thus, the first headline above says that too much or too little sleep (the cause) is “tied” to “ill health” (the effect). We also see that women should marry uglier men for a healthy marriage (in a study obviously written by an ugly man).
Now let us assume that there is a correlation that all these studies found. Let us assume that it is the case that people who sleep less than six hours a night weigh more than those who sleep eight hours a night, and that women who married uglier men (however that is defined) are in healthier (however that is defined) marriages. The fact of the matter is that when you compare any subset of a group, however you wish to define that subset, with the rest of the group as a whole, you will find things that the small group has in common at a statistically higher rate than the group as a whole. This happens automatically and does not mean that it is relevant in a causative sense!
To give a simple example, let’s examine hockey (since I like hockey). There are 30 teams in the NHL. Of those 30 teams, 7 are named after animals (the Penguins, Bruins, Thrashers, Panthers, Ducks, Coyotes, and Sharks) and 7 are named after people-groups (the Islanders, Rangers, Canadiens, Senators, Blackhawks, Oilers, and Kings). Each group of 7 constitutes 23% of the teams in the League.
There have been 80 Stanley Cups awarded since 1926. During that time, teams named after animals have won 8 Stanley Cups, which means that they won 10%. However, teams named after people-groups have won 39 Stanley Cups during that time, which means they won 49% of them. Clearly, having a team named after a people-group instead of after an animal provides a statistical advantage to a hockey team…
Perhaps someone could argue that the statistical data isn’t fair. After all, the Thrashers (1999), Panthers (1993), Ducks (1993), Coyotes (1996), and Sharks (1991) are all teams that did not exist before the 1990s! On the other hand, the Rangers, Canadiens, Senators, and Blackhawks all existed in 1926 (the start of this survey). Furthermore, the Kings were founded in 1967, the Oilers in 1971 and the Islanders in 1972. Of the animal teams, only the Bruins were around in 1926 (the Penguins were founded in 1967). Thus, using 1926 as the baseline (since before that there were other teams besides just NHL teams that could play for the Cup), the average year of founding for animal teams is 1981 and for people-group teams it’s 1945.
However, we can adjust for that. Animal teams have won a Cup on average every 3.25 years they’ve existed; while people-groups win a Cup for every 1.59 years they’ve existed. Clearly, it still remains better to have a team named after a people-group than an animal. (And I’m not biased since I cheer for the Avalanche, which is neither a people-group nor an animal…)
Now here’s the thing. The statistical data that I’ve given here is all correct (assuming I didn’t make any typos or anything of that nature), but every rational person would immediately recognize that the type of name a sports team has, has no bearing on the performance of that team. This is an attribute that is linked statistically, but the statistical linkage is accidental rather than causative.
Every time that we do these surveys and examine the numbers we have to realize that there are some number of things that will be discovered in common that are accidental correlations. The problem is that we ignore most of these connections. And when I say we ignore them, I don’t mean that we test the data and then go, “This isn’t relevant” but we do not even look for them in the first place. After all, were it not for the fact that I was looking for an example for this blog entry I would never have cared what percentage of teams named after animals won the Stanley Cup. This correlation would have been excluded a priori as being irrelevant.
But these irrelevant correlations are important to statistical analysis! Why? Because since a certain percentage of linkages are accidental, we have to account for them in our conclusion. In other words, we have to have some way of determining if the link we discover is causative or if it is merely the kind of statistical fluke you get when examining hockey mascots. And that means that we would need to examine all possible connections and discard those that are accidental in order to find out if the statistical percentages are covered.
That, however, is impractical to the point of impossibility. After all, it is relatively easy to come up with statistical correlations between things. For instance, with my hockey example it took me all of 15 minutes to come up with that correlation. The longest part was pulling up the Wiki sheets on the number of Stanley Cup wins various teams had had. Indeed, based on my experience I would argue that it is so easy to come up with meaningless links between data that it will always remain more likely that a correlation is accidental than causative. That is, for every one true causative link between a subset of a group and the average of the entire group, I would argue there are several accidental links. And these accidental links are not always as obviously accidental as the examples I’ve given. (For a less obvious example, think of the correlation between diabetes and obesity. Does one cause the other? Or is it just a statistical fluke, similar to the names of hockey teams?)
If it is so difficult to prove our position statistically due to the possibility of accidental links, then what good is it to come up with a statistical correlation in the first place? For most studies that you read about in the media, the answer is: “None.” However, for scientists there remains one thing that a truly causative link can do that an accidental link cannot do that saves the field. A truly causative link will enable you to make a prediction that you can test and verify. If something is causative then it will continue to cause the effect at the same rate. On the other hand, if it is accidental then it is a random linkage, and random linkages will break down through further testing. For instance, the fact that people-group teams have won more Stanley Cups than animal teams does not help us predict who will win the Stanley Cup this year or next year or the year after that; therefore, it is an accidental link rather than a causative link. However, if further testing shows that the percentages of obese people who get diabetes remains constant, then we can have more confidence that that is a truly causative link rather than simply a statistical accident.
So there are some ways to salvage statistics. But it requires that we be able to conduct further tests with our predictions in place in order to sort out whether we have a meaningful causative link or a meaningless accidental link. If we cannot conduct those further tests, then any causative links will be lost in the noise of the countless accidental links. They may be true, but it is impossible to verify it.
TOO MUCH, TOO LITTLE SLEEP TIED TO ILL HEALTH IN CDC STUDYAnd naturally we’re all aware of the competing studies that exist too. One study shows that eggs are bad for you; another that they’re good for you. One study shows how margarine is a healthier alternative than butter; another that butter is better for you. With so many competing studies, you can find a scientific backing for just about any position you want to take (especially in health matters).
Study: Long-Term Breast-Feeding Will Raise Child's IQ
WOMEN, WANT A HEALTHY MARRIAGE? MARRY MAN UGLIER THAN YOU, STUDY SAYS
STUDY: FOOD IN MCDONALD'S WRAPPER TASTES BETTER TO KIDS
Study: 1 in 50 U.S. babies abused, neglected in 2006
The existence of so many studies helps to emphasize a point regarding statistical analysis. Despite being a powerful tool, if you do not set up the guidelines and restrictions for your samples properly any statistics you observe won’t amount to a hill of beans. And we’re not even talking about the inherent fluctuations that require the existence of error bars (that’s the line that says +/- 3%, for example). Nor are we even addressing political manipulation of statistics in the form of pollaganda. Instead, I’m talking about something at the heart of statistics itself—it’s a universal.
To demonstrate what it is, let us first ask a simple question. When we do a statistical analysis of some observation, for what reason are we doing it? As you can see in the above headline examples, most of the time studies are done to find a causal linking between some object and/or action and some result. Thus, the first headline above says that too much or too little sleep (the cause) is “tied” to “ill health” (the effect). We also see that women should marry uglier men for a healthy marriage (in a study obviously written by an ugly man).
Now let us assume that there is a correlation that all these studies found. Let us assume that it is the case that people who sleep less than six hours a night weigh more than those who sleep eight hours a night, and that women who married uglier men (however that is defined) are in healthier (however that is defined) marriages. The fact of the matter is that when you compare any subset of a group, however you wish to define that subset, with the rest of the group as a whole, you will find things that the small group has in common at a statistically higher rate than the group as a whole. This happens automatically and does not mean that it is relevant in a causative sense!
To give a simple example, let’s examine hockey (since I like hockey). There are 30 teams in the NHL. Of those 30 teams, 7 are named after animals (the Penguins, Bruins, Thrashers, Panthers, Ducks, Coyotes, and Sharks) and 7 are named after people-groups (the Islanders, Rangers, Canadiens, Senators, Blackhawks, Oilers, and Kings). Each group of 7 constitutes 23% of the teams in the League.
There have been 80 Stanley Cups awarded since 1926. During that time, teams named after animals have won 8 Stanley Cups, which means that they won 10%. However, teams named after people-groups have won 39 Stanley Cups during that time, which means they won 49% of them. Clearly, having a team named after a people-group instead of after an animal provides a statistical advantage to a hockey team…
Perhaps someone could argue that the statistical data isn’t fair. After all, the Thrashers (1999), Panthers (1993), Ducks (1993), Coyotes (1996), and Sharks (1991) are all teams that did not exist before the 1990s! On the other hand, the Rangers, Canadiens, Senators, and Blackhawks all existed in 1926 (the start of this survey). Furthermore, the Kings were founded in 1967, the Oilers in 1971 and the Islanders in 1972. Of the animal teams, only the Bruins were around in 1926 (the Penguins were founded in 1967). Thus, using 1926 as the baseline (since before that there were other teams besides just NHL teams that could play for the Cup), the average year of founding for animal teams is 1981 and for people-group teams it’s 1945.
However, we can adjust for that. Animal teams have won a Cup on average every 3.25 years they’ve existed; while people-groups win a Cup for every 1.59 years they’ve existed. Clearly, it still remains better to have a team named after a people-group than an animal. (And I’m not biased since I cheer for the Avalanche, which is neither a people-group nor an animal…)
Now here’s the thing. The statistical data that I’ve given here is all correct (assuming I didn’t make any typos or anything of that nature), but every rational person would immediately recognize that the type of name a sports team has, has no bearing on the performance of that team. This is an attribute that is linked statistically, but the statistical linkage is accidental rather than causative.
Every time that we do these surveys and examine the numbers we have to realize that there are some number of things that will be discovered in common that are accidental correlations. The problem is that we ignore most of these connections. And when I say we ignore them, I don’t mean that we test the data and then go, “This isn’t relevant” but we do not even look for them in the first place. After all, were it not for the fact that I was looking for an example for this blog entry I would never have cared what percentage of teams named after animals won the Stanley Cup. This correlation would have been excluded a priori as being irrelevant.
But these irrelevant correlations are important to statistical analysis! Why? Because since a certain percentage of linkages are accidental, we have to account for them in our conclusion. In other words, we have to have some way of determining if the link we discover is causative or if it is merely the kind of statistical fluke you get when examining hockey mascots. And that means that we would need to examine all possible connections and discard those that are accidental in order to find out if the statistical percentages are covered.
That, however, is impractical to the point of impossibility. After all, it is relatively easy to come up with statistical correlations between things. For instance, with my hockey example it took me all of 15 minutes to come up with that correlation. The longest part was pulling up the Wiki sheets on the number of Stanley Cup wins various teams had had. Indeed, based on my experience I would argue that it is so easy to come up with meaningless links between data that it will always remain more likely that a correlation is accidental than causative. That is, for every one true causative link between a subset of a group and the average of the entire group, I would argue there are several accidental links. And these accidental links are not always as obviously accidental as the examples I’ve given. (For a less obvious example, think of the correlation between diabetes and obesity. Does one cause the other? Or is it just a statistical fluke, similar to the names of hockey teams?)
If it is so difficult to prove our position statistically due to the possibility of accidental links, then what good is it to come up with a statistical correlation in the first place? For most studies that you read about in the media, the answer is: “None.” However, for scientists there remains one thing that a truly causative link can do that an accidental link cannot do that saves the field. A truly causative link will enable you to make a prediction that you can test and verify. If something is causative then it will continue to cause the effect at the same rate. On the other hand, if it is accidental then it is a random linkage, and random linkages will break down through further testing. For instance, the fact that people-group teams have won more Stanley Cups than animal teams does not help us predict who will win the Stanley Cup this year or next year or the year after that; therefore, it is an accidental link rather than a causative link. However, if further testing shows that the percentages of obese people who get diabetes remains constant, then we can have more confidence that that is a truly causative link rather than simply a statistical accident.
So there are some ways to salvage statistics. But it requires that we be able to conduct further tests with our predictions in place in order to sort out whether we have a meaningful causative link or a meaningless accidental link. If we cannot conduct those further tests, then any causative links will be lost in the noise of the countless accidental links. They may be true, but it is impossible to verify it.
Sunday, May 04, 2008
The Three Prisoners Problem
One thing that can be both a blessing and a curse about my nature is that I am often able to find ways to keep myself awake almost all night long. It doesn’t matter how tired I am when I try to go to bed, if I think of something that gets my mind going then I’d much rather continue to think on it than sleep. This happened to me the other day as I was thinking of a few statistical quirks regarding Natural Selection, random mutations, and the like.
Unfortunately, I’m not yet able to write the blog post that I wanted to write, because while I know exactly what I’m talking about, it lacks sufficient groundwork for many other readers to be able to follow along! Since I am an apologist at heart (one who would love to preside over the complete destruction of ideological Darwinism, mind you) I do wish to expand on my thoughts and present them to others, so this leaves me with the necessary task of providing some starting groundwork before I get to the main point. And besides, although it’s tangential to my ultimate point, some of this stuff is just plain kewl :-)
In any case, since a great deal of what I will be focusing on in future posts will deal with statistical analysis, I thought it might be beneficial to give a quick overview of The Three Prisoners Problem in order to A) melt your brain if you’ve never heard of it and B) show how statistics can be logical and yet make no sense at first glance (mostly due to a wrong perspective).
The Three Prisoners Problem was originally mentioned by Martin Gardner in his “Mathematical Games” column in the October, 1959 edition of Scientific American, but under the Monty Hall guise (its mathematical equivalent) it has gathered more infamy, especially after Marilyn vos Savant’s article in Parade magazine in 1990. If this doesn’t make sense to you at first glance, you can take comfort in the fact that it has fooled Nobel laureates, professional mathematicians, and Mensa members countless times. Here I will give my own version of the problem.
There are three prisoners in the king’s dungeon: Adam, Bill, and Charlie. The Warden arrives at each cell and says, “The King has decided that two of you shall go free tomorrow.” At this, there is great rejoicing. But the Warden continues: “However, one of you will be executed.”
“Who will it be?” they all ask in turn.
The Warden responds: “The King has told me who will be executed, but he has also forbidden me telling you who will live and who will die.”
Each of the prisoners accepts this answer except for Charlie. Charlie is a shrewd character and because he knows the Warden is scrupulously honest, he asks: “I know you said that you cannot tell me who will be executed or who will be set free, and therefore you cannot tell me my fate. But will you instead give me one name of one of the other prisoners who will be set free?”
The Warden thinks about this for a moment. “Why would you want to know that?” he ponders. “If I don’t give you a name, you know that you have a 1/3 chance of being executed and a 2/3 chance of going free. If I tell you a name, then you will only have a 1/2 chance of going free! It is better for you if you do not know a name.”
“In that case,” Charlie responds, “why not tell me?”
The Warden relents and says, “Adam will go free tomorrow.”
At this, Charlie sits back and smiles because the Warden has inadvertently told him that it is twice as likely that Bill will be executed as it is that he will be executed…
The reason this is a “problem” is because for most of us we reason the way that the Warden did. Surely telling Charlie that Adam will go free has actually reduced Charlie’s odds of survival, hasn’t it? It used to be 2/3 because it could have been Adam, Bill, or Charlie who would be killed and 2 of them would have lived. But now it’s either Bill or Charlie who will be killed and only one of them would live, and that’s a 1/2 chance, isn’t it?
There are two ways to look at this. First, let’s look at the mathematical rule involved: fractional statistics must together add up to 1.
When the prisoners are first given information, there is a 1/3 chance for each of them that they will be killed. Thus, we have the odds of death being:
Adam = 1/3
Bill = 1/3
Charlie = 1/3
Now when Charlie asks which of the first two prisoners will go free, since the Warden is honest, he tells him that Adam is one who will go free. But this gives no new information to Charlie about whether or not Charlie will die. Charlie’s odds of being killed remain 1/3. However, Adam’s odds of being killed are reduced to 0. He will survive.
If Charlie has a 1/3 shot of dying and Adam has a 0 shot of dying, then because statistics must balance to 1 (it is a certainty that someone will die), this means that Bill’s odds of dying must be 2/3. As a result, Bill is twice as likely to be executed as Charlie.
Of course, this still doesn’t seem right at all! After all, how can telling Charlie that Adam will go free affect Bill’s odds of survival but not affect Charlie’s original odds of survival?
The second way of explaining this helps to flesh it out a bit better. As we stated, when the problem begins, each prisoner has a 1/3 chance of being killed. Therefore, there are three possible options. Let us examine these three options and what the Warden must respond under each option.
Option 1: Adam is killed. If Adam is the one to be executed, then when Charlie asks for the name of one of the two prisoners who will live, the Warden must respond “Bill.” If he says Adam lives, then he has lied (and we’ve stipulated that the Warden is honest). Conclusion: Charlie lives; the prisoner not named dies.
Option 2: Bill is killed. Like the above, the Warden’s choice is restricted to one answer. The Warden can only say that “Adam” will live. Conclusion: Charlie lives; the prisoner not named dies.
Option 3: Charlie is killed. Here is the only instance where the Warden has freedom. Since Charlie will be killed, then he can name either Adam or Bill. Conclusion: Charlie dies; the prisoner not named lives.
As we see in the above, Charlie’s chances of being killed remain 1/3 because only under option 3 does he die. Further, 2/3 of the time the Warden is forced to name a specific prisoner because the one not named is the one who will die. Therefore, 2/3 of the time the prisoner not named is the prisoner who will be executed.
This is also easier to see if we use bigger numbers. Suppose that there are instead 1,000 prisoners and all but one of them will be set free while the remaining prisoner is executed. Under these circumstances, the Warden reveals 998 prisoners who will be set free, leaving only Charlie and prisoner number 473 behind. Which is more likely, that the Warden was forced to leave prisoner number 473 as an option or that Charlie is going to be killed and prisoner 473 was a random selection? Obviously, there is only a 1/1000 chance that prisoner 473 was a random selection, but there is a 999/1000 chance that prisoner 473 was the forced choice. So in this case, the reason it is counterintuitive has more to do with the fact that we do not realize the Warden is excluding all but one prisoner from his answer. If there were 1,000 prisoners total and Charlie asked for the list of 998 of them that would go free, the Warden would immediately spot this error.
Note, however, that even under the circumstance that Charlie only asked for the name of one prisoner out of the 1000 who would go free, that would decrease the odds of all the other unnamed prisoners surviving, although in this instance the amount the odds change would be negligible. Charlie would remain with a 1/1000 chance of dying, while the 998 unnamed prisoners would have just over a 1.001/1000 chance of dying and the one named one would have a no (0) chance of dying. This equates to 998 prisoners splitting a 999/1000 odds, so you still end up with 1/1000 + 999/1000 + 0 = 1. (1.001 x 998 rounds to 999.)
As I mentioned at the top of this post, this is mathematically equivalent to the Monty Hall Problem. That can be demonstrated while keeping with the prisoner motif in the following manner. Suppose that instead of the Warden talking to the prisoners, the King summons the Warden to his throne room. The King, who enjoys tormenting the Warden, says:
“Warden, I am going to execute two prisoners tomorrow, but I am going to free one of them. I have written his name down and locked it in this chest beside me along with one thousand gold pieces. If you can guess who will go free, you can have all the gold in the chest. If you do not guess who goes free, you will have to join the prisoners being executed!”
The Warden realizes he has a 1/3 chance of gaining riches and a 2/3 chance of dying. Nevertheless, the King has given him no option. So he says, “I pick Adam to live.”
The King smiles and says, “Let us make this more interesting. Before you open the chest and see the name, I will tell you that Charlie is going to die. Now, do you still want to choose Adam to live, or do you want to switch your choice to Bill?”
At this point, what should the Warden decide?
Again, mathematically this is equivalent to the Three Prisoners Problem above. Therefore, we know that when the Warden picked Adam to live, he had a 1/3 chance of being right. The King has now informed the Warden that Charlie will die: therefore, Charlie has a 0 chance of living. Once again, because the numbers have to add up to 1, this means that Bill now has a 2/3 chance of living and Adam only has a 1/3 chance of living. Therefore, the Warden should switch his choice.
And to demonstrate this in the similar manner as above, look at the three options of what would happen after the Warden picks Adam but before the King (who already know who will die) responds:
Option 1: Adam lives. In this case, the King can name either Bill or Charlie as dying. Therefore, the Warden should not switch his choice because whomever the King does not name of the other two prisoners will die.
Option 2: Bill lives. In this case, the King MUST name Charlie as dying. The Warden should change his pick to the prisoner not mentioned (Bill).
Option 3: Charlie lives. In this case, the King MUST name Bill as dying. The Warden should change his pick to the prisoner not mentioned (Charlie).
Again we see that 2/3 of the time, the Warden should change his selection.
So here we see that sometimes statistics can be perfectly logical and rational, yet the result is so counterintuitive that they feel wrong. In my next post, I’ll give an example of the opposite: when statistics are irrational and yet seem to make sense. After that, I will look at a few examples statistics in action with Darwinism.
Unfortunately, I’m not yet able to write the blog post that I wanted to write, because while I know exactly what I’m talking about, it lacks sufficient groundwork for many other readers to be able to follow along! Since I am an apologist at heart (one who would love to preside over the complete destruction of ideological Darwinism, mind you) I do wish to expand on my thoughts and present them to others, so this leaves me with the necessary task of providing some starting groundwork before I get to the main point. And besides, although it’s tangential to my ultimate point, some of this stuff is just plain kewl :-)
In any case, since a great deal of what I will be focusing on in future posts will deal with statistical analysis, I thought it might be beneficial to give a quick overview of The Three Prisoners Problem in order to A) melt your brain if you’ve never heard of it and B) show how statistics can be logical and yet make no sense at first glance (mostly due to a wrong perspective).
The Three Prisoners Problem was originally mentioned by Martin Gardner in his “Mathematical Games” column in the October, 1959 edition of Scientific American, but under the Monty Hall guise (its mathematical equivalent) it has gathered more infamy, especially after Marilyn vos Savant’s article in Parade magazine in 1990. If this doesn’t make sense to you at first glance, you can take comfort in the fact that it has fooled Nobel laureates, professional mathematicians, and Mensa members countless times. Here I will give my own version of the problem.
There are three prisoners in the king’s dungeon: Adam, Bill, and Charlie. The Warden arrives at each cell and says, “The King has decided that two of you shall go free tomorrow.” At this, there is great rejoicing. But the Warden continues: “However, one of you will be executed.”
“Who will it be?” they all ask in turn.
The Warden responds: “The King has told me who will be executed, but he has also forbidden me telling you who will live and who will die.”
Each of the prisoners accepts this answer except for Charlie. Charlie is a shrewd character and because he knows the Warden is scrupulously honest, he asks: “I know you said that you cannot tell me who will be executed or who will be set free, and therefore you cannot tell me my fate. But will you instead give me one name of one of the other prisoners who will be set free?”
The Warden thinks about this for a moment. “Why would you want to know that?” he ponders. “If I don’t give you a name, you know that you have a 1/3 chance of being executed and a 2/3 chance of going free. If I tell you a name, then you will only have a 1/2 chance of going free! It is better for you if you do not know a name.”
“In that case,” Charlie responds, “why not tell me?”
The Warden relents and says, “Adam will go free tomorrow.”
At this, Charlie sits back and smiles because the Warden has inadvertently told him that it is twice as likely that Bill will be executed as it is that he will be executed…
The reason this is a “problem” is because for most of us we reason the way that the Warden did. Surely telling Charlie that Adam will go free has actually reduced Charlie’s odds of survival, hasn’t it? It used to be 2/3 because it could have been Adam, Bill, or Charlie who would be killed and 2 of them would have lived. But now it’s either Bill or Charlie who will be killed and only one of them would live, and that’s a 1/2 chance, isn’t it?
There are two ways to look at this. First, let’s look at the mathematical rule involved: fractional statistics must together add up to 1.
When the prisoners are first given information, there is a 1/3 chance for each of them that they will be killed. Thus, we have the odds of death being:
Adam = 1/3
Bill = 1/3
Charlie = 1/3
Now when Charlie asks which of the first two prisoners will go free, since the Warden is honest, he tells him that Adam is one who will go free. But this gives no new information to Charlie about whether or not Charlie will die. Charlie’s odds of being killed remain 1/3. However, Adam’s odds of being killed are reduced to 0. He will survive.
If Charlie has a 1/3 shot of dying and Adam has a 0 shot of dying, then because statistics must balance to 1 (it is a certainty that someone will die), this means that Bill’s odds of dying must be 2/3. As a result, Bill is twice as likely to be executed as Charlie.
Of course, this still doesn’t seem right at all! After all, how can telling Charlie that Adam will go free affect Bill’s odds of survival but not affect Charlie’s original odds of survival?
The second way of explaining this helps to flesh it out a bit better. As we stated, when the problem begins, each prisoner has a 1/3 chance of being killed. Therefore, there are three possible options. Let us examine these three options and what the Warden must respond under each option.
Option 1: Adam is killed. If Adam is the one to be executed, then when Charlie asks for the name of one of the two prisoners who will live, the Warden must respond “Bill.” If he says Adam lives, then he has lied (and we’ve stipulated that the Warden is honest). Conclusion: Charlie lives; the prisoner not named dies.
Option 2: Bill is killed. Like the above, the Warden’s choice is restricted to one answer. The Warden can only say that “Adam” will live. Conclusion: Charlie lives; the prisoner not named dies.
Option 3: Charlie is killed. Here is the only instance where the Warden has freedom. Since Charlie will be killed, then he can name either Adam or Bill. Conclusion: Charlie dies; the prisoner not named lives.
As we see in the above, Charlie’s chances of being killed remain 1/3 because only under option 3 does he die. Further, 2/3 of the time the Warden is forced to name a specific prisoner because the one not named is the one who will die. Therefore, 2/3 of the time the prisoner not named is the prisoner who will be executed.
This is also easier to see if we use bigger numbers. Suppose that there are instead 1,000 prisoners and all but one of them will be set free while the remaining prisoner is executed. Under these circumstances, the Warden reveals 998 prisoners who will be set free, leaving only Charlie and prisoner number 473 behind. Which is more likely, that the Warden was forced to leave prisoner number 473 as an option or that Charlie is going to be killed and prisoner 473 was a random selection? Obviously, there is only a 1/1000 chance that prisoner 473 was a random selection, but there is a 999/1000 chance that prisoner 473 was the forced choice. So in this case, the reason it is counterintuitive has more to do with the fact that we do not realize the Warden is excluding all but one prisoner from his answer. If there were 1,000 prisoners total and Charlie asked for the list of 998 of them that would go free, the Warden would immediately spot this error.
Note, however, that even under the circumstance that Charlie only asked for the name of one prisoner out of the 1000 who would go free, that would decrease the odds of all the other unnamed prisoners surviving, although in this instance the amount the odds change would be negligible. Charlie would remain with a 1/1000 chance of dying, while the 998 unnamed prisoners would have just over a 1.001/1000 chance of dying and the one named one would have a no (0) chance of dying. This equates to 998 prisoners splitting a 999/1000 odds, so you still end up with 1/1000 + 999/1000 + 0 = 1. (1.001 x 998 rounds to 999.)
As I mentioned at the top of this post, this is mathematically equivalent to the Monty Hall Problem. That can be demonstrated while keeping with the prisoner motif in the following manner. Suppose that instead of the Warden talking to the prisoners, the King summons the Warden to his throne room. The King, who enjoys tormenting the Warden, says:
“Warden, I am going to execute two prisoners tomorrow, but I am going to free one of them. I have written his name down and locked it in this chest beside me along with one thousand gold pieces. If you can guess who will go free, you can have all the gold in the chest. If you do not guess who goes free, you will have to join the prisoners being executed!”
The Warden realizes he has a 1/3 chance of gaining riches and a 2/3 chance of dying. Nevertheless, the King has given him no option. So he says, “I pick Adam to live.”
The King smiles and says, “Let us make this more interesting. Before you open the chest and see the name, I will tell you that Charlie is going to die. Now, do you still want to choose Adam to live, or do you want to switch your choice to Bill?”
At this point, what should the Warden decide?
Again, mathematically this is equivalent to the Three Prisoners Problem above. Therefore, we know that when the Warden picked Adam to live, he had a 1/3 chance of being right. The King has now informed the Warden that Charlie will die: therefore, Charlie has a 0 chance of living. Once again, because the numbers have to add up to 1, this means that Bill now has a 2/3 chance of living and Adam only has a 1/3 chance of living. Therefore, the Warden should switch his choice.
And to demonstrate this in the similar manner as above, look at the three options of what would happen after the Warden picks Adam but before the King (who already know who will die) responds:
Option 1: Adam lives. In this case, the King can name either Bill or Charlie as dying. Therefore, the Warden should not switch his choice because whomever the King does not name of the other two prisoners will die.
Option 2: Bill lives. In this case, the King MUST name Charlie as dying. The Warden should change his pick to the prisoner not mentioned (Bill).
Option 3: Charlie lives. In this case, the King MUST name Bill as dying. The Warden should change his pick to the prisoner not mentioned (Charlie).
Again we see that 2/3 of the time, the Warden should change his selection.
So here we see that sometimes statistics can be perfectly logical and rational, yet the result is so counterintuitive that they feel wrong. In my next post, I’ll give an example of the opposite: when statistics are irrational and yet seem to make sense. After that, I will look at a few examples statistics in action with Darwinism.
Sunday, September 17, 2006
Priorities And What's "Not Surprising"
"What Pew actually did over two weeks in May was ask 820 self-identifying American Christians 'Do you think of yourself first as American or as Christian?'...Not surprisingly, the 'Christian first' response emanated disproportionately from self-identified Evangelicals, 62% of whom said 'Christian first.' By contrast, the figures for other major Christian sectors were nearly reversed, with 62% of Catholics and 65% of Mainline Protestants saying 'American first'....One conclusion you might draw from the apparent willingness to go eeny, meeny with one's sympathies is that the separation of church and state is alive and well. All you liberals who worry that you live in an age when Christianity and patriotism have become inextricably intertwined? You can stop worrying. Most Americans polled could not only distinguish church from state, but were quite comfortable explaining where their primary allegiance lay." (Time)
Notice how Time's David Van Biema uses the words "Not surprisingly". There's a lot that's good about Evangelicalism. There are a lot of problems in Evangelicalism as well, but other groups have worse problems.
Albert Mohler discussed this topic on his radio program last Friday.
Notice how Time's David Van Biema uses the words "Not surprisingly". There's a lot that's good about Evangelicalism. There are a lot of problems in Evangelicalism as well, but other groups have worse problems.
Albert Mohler discussed this topic on his radio program last Friday.
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