Math Help - The role of prior knowledge in probability.

In his book ''PARADOX'' Professor Jim Al-Khalili poses the following question ( I have shortened it somewhat for brevity): You wish to buy two kittens. The pet shop has two sibling kittens, one black and one tabby. You ask if they are boy or girl and you have to consider two different responses from the pet shop owner.
a) She tells you: 'I have only checked one of them and it's a boy.'
b) She tells you: 'I have checked the tabby and it's a boy.'
The question to be answered is: What is the probability that they are both boys?
After giving some explanation, the professor concludes by writing: So you see the probobility that both kittens are boys changes from 1 in 3 to 1 in 2 as soon as you know which of the two cats has been discovered to be a boy.
My problem is that I can't grasp why the probability should change. After all, once you are told one of the kittens is a boy why should it matter to the probability of both being boys whether it is the black one or the tabby? Would someone care to explain where my understanding is wrong ?
P.S The professor also writes that this is the same situation as that in the Monty Hall Paradox. No doubt those who read this will be familiar with this old TV show conundrum.

Think about the four "equally likely" possibilities:
1) Both are males
2) The tabby kitten is a male and the black kitten is a female
3) The black kitten is a male and the tabby kitten is a female
4) Both are females

Which of those does "at least one is a male" exclude?

Which of those does "the tabby is a male" exclude?

Thank you HallsofIvy. Your explanation and the professor's agree. However, I think my problem is this. The question asked is : What is the probability both are boys? I reason that there are, in fact. only three possibilities:-
1) Both are boys ( I suppose males would be a better description but never mind).
2) Both are girls.
3) One is a boy and one a girl.
When we are told one is a boy (2) is ruled out. The probability both are boys is then 1 in 2.
By giving descriptions to the two kittens and, by so doing, make it four possibilities is, to my way of thinking, illogical.
Am I still missing something ? (Please don't tell me I'm missing a sandwich from my loaf or I shall be hurt).

Originally Posted by gordonX

Thank you HallsofIvy. Your explanation and the professor's agree. However, I think my problem is this. The question asked is : What is the probability both are boys? I reason that there are, in fact. only three possibilities:-
1) Both are boys ( I suppose males would be a better description but never mind).
2) Both are girls.
3) One is a boy and one a girl.

You can write it like that but then these are no longer "equally likely". Assuming for simplicity that "boys" and "girls' are equally likely, then the probability or two boys is (1/2)(1/2)= 1/4. The probability of two girls is (1/2)(1/2)= 1/4, leaving a probability of 1/2 for "one boy and one girl". Instead, we can say that the probability of a specific one of the kittens, say the tabby, being a boy and the other a girl is (1/2)(1/2)= 1/4 so that the probability of that specific kitten being a boy and the other a girl is also (1/2)(1/2)= 1/4. Putting those together, we have the probability of one being a girl and the other a boy is 1/4+ 1/4= 1/2.

You are ignoring the importance of comparing only equally likely cases

When we are told one is a boy (2) is ruled out. The probability both are boys is then 1 in 2.
By giving descriptions to the two kittens and, by so doing, make it four possibilities is, to my way of thinking, illogical.
Am I still missing something ? (Please don't tell me I'm missing a sandwich from my loaf or I shall be hurt).

No, only, as I said, that we have to compare "equally likely" events.

GordonX, you have a case of justifiable confusion. Let's start from the beginning. Suppose someone tells you that one of a pair of kittens is a boy and the other is unknown (as in the book Paradox, pages 18-19) and you want to know the odds that both are boys. There are four equally likely possibilities: BOY-BOY, BOY-GIRL, GIRL-BOY, GIRL-GIRL. This maps to the case of the four equally likely possibilities when you flip two coins: 1/4 chance for HEADS-HEADS, 1/4 chance for HEADS-TAILS, 1/4 chance for TAILS-HEADS, 1/4 chance for TAILS-TAILS. If you learn that one kitten is male, it’s like learning that one coin toss landed heads. (I imagine coins more readily than kittens. Please bear with me.)

Now suppose a machine (or a friend) flips two coins, randomly picks one of the coins and tells you whether it’s heads or tails. Let’s say that it’s heads. According to Paradox, you should now think that there is a one in three chance that both coin tosses landed heads and a two in three chance that one is heads and the other tails. Similarly, if the machine says tails, you should now think that there is a one in three chance that both coin tosses landed tails and a two in three chance that one is heads and the other tails. Let us add the probabilities, according to Paradox, of when the machine says heads and when it says tails. When the machine says heads, this yields odds of: 1/3 heads and 2/3 heads + tails (in either order). When the machine says tails, this yields odds of: 1/3 tails and 2/3 heads + tails (in either order). Half the time the machine will say heads and half the time it will say tails, so we must multiply each set of probabilities by a half and then add them together. This yields: 1/6 heads; 2/3 heads + tails (in either order); 1/6 tails. This fails to match the odds given in the first paragraph, so Paradox is wrong.

When our machine announces heads, it has picked one of four (not three) heads in the possibilities list: two in HEADS-HEADS, one in HEADS-TAILS and one in TAILS-HEADS. We may list the possibilities as: HEADS-HEADS, the other HEADS-HEADS, HEADS-TAILS, TAILS-HEADS. The odds that the unknown coin toss will be heads is 1/2. Similarly, when the machine announces tails, it has picked one of four (not three) tails in the possibilities list. When we sum these probabilities, they match those in the first paragraph.

An intuitive approach supports this. Flip two coins (these may be black, tabby, or whatever), let them land on a table, pick one at random, announce whether it is heads or tails, set it aside and intuit the odds of the second coin being heads or tails. See why it’s easier to think in terms of coins? (It’s harder to flip cat genitals.)
I emailed Jim Al-Khalili, the author of Paradox, but he didn’t respond.

Did you actually do that experiment and get those results? I suspect you didn't and your reasoning is incorrect.

Dear gordonX, I read the corresponding section in Jim Al-Khalili's book. Although the formulation of the paradox is exactly as you have posted, in his explanation instead of saying "I have only checked one of them and it's a boy" (call it phrase 1), Al-Khalili is using the phrase "At least one of them is a boy" (call it phrase 2). The explanation and the reasoning presented in his book is valid when you consider phrase 2. However, it is not valid if you consider the phrase 1. Let me explain why the two phrases are not equivalent:
When you are informed by phrase 1 or 2 about the gender of one of the kittens, the possible outcomes are three a) boy-boy, b) boy-girl, c) girl-boy. Al-Khalili argues that you should assign equal probabilities to each outcome, and this is the case if you are based on information of phrase 2. However, in case of phrase 1 you can use additional information, since you know that only one of the kittens was examined and it was a boy. In case of a) this is expected with probability 1, whereas in case of b) or c) it is expected with probability 1/2. If you normalize the above values to add to 1, you should assign a) with 1/2, b) with 1/4 and c) with 1/4. Then the probability of being both boys is 1/2 like in the case you are told "I have checked the tabby and it's a boy". Al-Khalili's point about prior knowledge can be made based on phrase 2, but using phrase 1 as equivalent was a mistake.

Dear GoodListener, I hope you found as much of interest in the book as I did. I get a great deal of pleasure in reading such books, and I particularly like reading of the mysteries of prime numbers. I hasten to add I make no pretence of being a mathematician, which is obvious from the difficulty I am having with this puzzle. I can see that we always start with four possibilities ie B-B, G-G, B-G, G-B. This is the same as tossing two coins is it not when the possible outcomes are H-H, T-T, H-T, T-H. However, when you are told that one cat is a B it seems to me (and this is where I lose the plot because I cannot see that it matters which one is identified as a B) that there can now only be two possibilities for the other; B or G. Hence the probability of B-B is 1/2 and never 1/3. What the heck is the difference in saying "I've only checked one and it's a B" and "I've checked tabby and he's a B"? The gender of one is now certain and that only leaves the gender of the other to be matched which can only be a 50/50 guess.
It's a great life if you don't weaken somebody once said.

Originally Posted by NORM

GordonX, you have a case of justifiable confusion. Let's start from the beginning. Suppose someone tells you that one of a pair of kittens is a boy and the other is unknown (as in the book Paradox, pages 18-19) and you want to know the odds that both are boys. There are four equally likely possibilities: BOY-BOY, BOY-GIRL, GIRL-BOY, GIRL-GIRL. This maps to the case of the four equally likely possibilities when you flip two coins: 1/4 chance for HEADS-HEADS, 1/4 chance for HEADS-TAILS, 1/4 chance for TAILS-HEADS, 1/4 chance for TAILS-TAILS. If you learn that one kitten is male, it’s like learning that one coin toss landed heads. (I imagine coins more readily than kittens. Please bear with me.)

Now suppose a machine (or a friend) flips two coins, randomly picks one of the coins and tells you whether it’s heads or tails. Let’s say that it’s heads. According to Paradox, you should now think that there is a one in three chance that both coin tosses landed heads and a two in three chance that one is heads and the other tails. Similarly, if the machine says tails, you should now think that there is a one in three chance that both coin tosses landed tails and a two in three chance that one is heads and the other tails. Let us add the probabilities, according to Paradox, of when the machine says heads and when it says tails. When the machine says heads, this yields odds of: 1/3 heads and 2/3 heads + tails (in either order). When the machine says tails, this yields odds of: 1/3 tails and 2/3 heads + tails (in either order). Half the time the machine will say heads and half the time it will say tails, so we must multiply each set of probabilities by a half and then add them together. This yields: 1/6 heads; 2/3 heads + tails (in either order); 1/6 tails. This fails to match the odds given in the first paragraph, so Paradox is wrong.

When our machine announces heads, it has picked one of four (not three) heads in the possibilities list: two in HEADS-HEADS, one in HEADS-TAILS and one in TAILS-HEADS. We may list the possibilities as: HEADS-HEADS, the other HEADS-HEADS, HEADS-TAILS, TAILS-HEADS. The odds that the unknown coin toss will be heads is 1/2. Similarly, when the machine announces tails, it has picked one of four (not three) tails in the possibilities list. When we sum these probabilities, they match those in the first paragraph.

An intuitive approach supports this. Flip two coins (these may be black, tabby, or whatever), let them land on a table, pick one at random, announce whether it is heads or tails, set it aside and intuit the odds of the second coin being heads or tails. See why it’s easier to think in terms of coins? (It’s harder to flip cat genitals.)
I emailed Jim Al-Khalili, the author of Paradox, but he didn’t respond.

Not only are you wrong, but you should not use the term "odds" without defining what you mean because it means different things to different communities, better yet why not use the term probability since it is unambiguous.

.

Dear gordonX, I have selectively read some of the chapters, since I was familiar with most of the presented material. It is a very interesting book. Regarding the prior knowledge in probability part, I will have to insist that as I argue in my previous post Al-Khalili's phrasing is wrong he should use (as he does in his explanation) the phrase "At least one of them is a boy" . Then the probabilities are 1/3 for each of the possible outcomes b-b, b-g, g-b. If you erroneously use the phrase "I have only checked one of them and it's a boy" the three outcomes should not be considered equiprobable and the result is the same as with the "I have checked the tabby and it's a boy" case.
Best regards...

Originally Posted by GoodListener

Dear gordonX, I have selectively read some of the chapters, since I was familiar with most of the presented material. It is a very interesting book. Regarding the prior knowledge in probability part, I will have to insist that as I argue in my previous post Al-Khalili's phrasing is wrong he should use (as he does in his explanation) the phrase "At least one of them is a boy" . Then the probabilities are 1/3 for each of the possible outcomes b-b, b-g, g-b. If you erroneously use the phrase "I have only checked one of them and it's a boy" the three outcomes should not be considered equiprobable and the result is the same as with the "I have checked the tabby and it's a boy" case.
Best regards...

Try the following experiments (or if you know how you could simulate it)

1. Take two coins and colour one blue and the other red.

Take another coin and colour one side blue and the other red.

Set the count N1 of number of valid cases to zero, set the count N2 of both heads and valid to zero.

Reapeat the following at least 200 times:
Toss all three coins, if the bi-colour coin showns blue examine the blue coin, if red check the red coin, if the checked coin is heads we are on and add one to N1 the count of valid cases. Now examine the other coin, if it is heads add one to N2 the count of both heads and valid.

Now what is p=N2/N1?

The point of this? It is the same as if we had a specified one being heads

2. Now repeat but instead of using the bi-colour coin to chose which of the others to check just always check the red coin first, if heads increment N1 by 1, then check the other coin ...

The point of this? It is the case where a specified one is heads.

Both of these p should be equal to about 0.5.

3. Now if instead we just toss the two coins, and increment N1 if either are heads, and N2 if both are heads.

The point of this? It is the case where one or more is heads.

In this case p should be about 1/3.


The first two experiments correspond to being told the tabby/black kitten is a boy, and the third case to being told that at least one is a boy.

Case b as posted is supposd to correspond to experiment 3, though the wording is ambiguous and it might be interpreted as corresponding to experiment 1. Now what the wording in the book specifies I don't know.

Dear zzephod, the phrasing in the book is as gordonX quotes in the leading post, i.e.
"In his book ''PARADOX'' Professor Jim Al-Khalili poses the following question ( I have shortened it somewhat for brevity): You wish to buy two kittens. The pet shop has two sibling kittens, one black and one tabby. You ask if they are boy or girl and you have to consider two different responses from the pet shop owner.
a) She tells you: 'I have only checked one of them and it's a boy.'
b) She tells you: 'I have checked the tabby and it's a boy.'
The question to be answered is: What is the probability that they are both boys?"

Therefore, as I argue (and if I understand correctly your previous post as you also argue) p=0.5 for both cases. It is the third case, "At least one of them is a boy" that yields p=1/3. Part of the confusion is that in his explanation Al-Khalili uses the phrase "At least one of them is a boy" as if it is equivalent to "I have only checked one of them and it's a boy", which we both seem to agree that is not actually equivalent.
Best regards...

Dear All,
Having thought about all the replies I am still unable to come to any other conclusion than that the probability that both cats are male is 0.5. Whatever terminology the shop use, whether it be "I have only checked one of them and it's a boy" or "I have checked the tabby and it's a boy" or even if they had said "At least one of them is a boy" I now know that there is only the other cat to consider i.e is it boy or girl. I quite understand that if you don't know the sex of either then it is a one in four chance that both are boys, but once one of them is identified as a boy, and it matters not to my mind whether it is the black or tabby, then it is the sex of only one that is unknown.

Considering the two coin analogy, supposing I toss them both (call them coins A and B). I look at A (or B) and find it shows heads. I then say to you "What is the probability that the other coin shows heads". I think you will tell me the probability is 0.5. That being the case the probability that both coins show heads is also now 0.5.
Kind regards to all for trying to put me on the right track.

Dear gordonX, let me set some context regarding probabilities, their calculation and interpretation. Although there are several approaches to probabilities (subjective probabilities, frequentism, Bayesian inference), the most successful is the axiomatic approach of modern probability theory. A basic concept of this theory is the sample space of the possible outcomes and a measure of probability. According to this theory you are free to assign any probability values to the outcomes (as long as you are consistent with the axioms). Then, you can use the theory to calculate the probabilities of events which are subsets of the sample space. Most of the controversy around probability problems is about the first step of assigning probability values to outcomes. It all comes down to what assumptions you are using and how accurate is the description of the problem. Thus, in our case all three phrases are equivalent and result to p=1/2 only if you assume that in all cases only one of the kittens is checked. On the other hand, if you assume that hearing "At least one of them is a boy" means that someone checks both kittens and iff at least one is a boy he lets you know about it you are compelled to assign equal probabilities to b-b, b-g and g-b yielding p=1/3. When I say you are compelled, I mean by philosophical reasons like the principle of indifference, symmetry etc (but not by modern probability theory which does not examine how the initial values are derived). I hope this explanation is helpful.
Best regards...

Dear zzephod, following my previous post I have to comment that uniformity is a popular assumption. However, it is often overused. For example in the experiment you propose you suggest using a bicolor coin (resulting to uniform probabilities for checking the monocolored coins). There is no reason to assume that. You can check only the first coin with probability q and only the second coin with probability 1-q. In this case however it doesn't change the results of the experiment. No matter what value you assign to q, p is always 1/2.
Best regards...