# Coin tossing game

Suppose two people play a game. Person A tosses two (fair) coins, and person B tries to guess the outcome. Suppose furthermore than person A eliminates one of the four outcomes.

On a given toss, person A might say that the outcome is not TT. Then by eliminating that outcome and normalizing the probabilities, we would say that there is a 2/3 chance of getting a heads/tails pair on this toss.

On the other hand, suppose person A instead shows person B that one of the coins came up heads (but does not say whether this is the first or second coin). Then the probability of the unknown coin being tails is 1/2, so we would compute the probability of a heads/tails pair is 1/2. But showing that one coin is heads is the same as eliminating the TT option!

Obviously, both of these calculations cannot be right, but I cannot see the error in reasoning in either case. Which is correct, and why is the other one wrong?

• Commented Jun 9, 2015 at 19:21

I'm posting an answer here after reviewing the wikipedia article. That article has a much more in-depth discussion, but here is the short version, as I understand it.

Scenario 1 is correct. In scenario 2, however, there is a hidden ambiguity. Did person A choose the coin to show person B, or did they grab a coin at random? If person A picked the coin deliberately because it was heads, then A has only given B the information that the outcome was 'not TT.' The probability is therefore 2/3 because we can permute the coins as nosyarg said. Alternately if A randomly grabbed a coin to show us, then this act of randomly selecting one of the two coins introduces an assymmetry between the two coins, so we can no longer permute them. The probability is thus 1/2.

In other words, the probability will depend on the selection method for the 'shown coin.' As the wikipedia article discusses, this is the reason for the confusion: the question doesn't specify how the coin is selected, and depending on how we assume it was selected, we get different probabilities.

In the boy-girl version, the 'paradoxical' version again doesn't specify how the sibling is chosen. If we meet a person and find out they have a sibling, then the sibling's gender probability (assuming that both people fall into male or female genders) will depend on how we met the person:

1. If we met the person in a situation where we were equally likely to meet males and females, then the two people constitute a pair from which the 'met person' was randomly selected. So the gender of the second person gets a 1/2 chance either way.

2. If we met the person in a situation where only one gender can be present, then we get a 2/3 chance of the sibling having the opposite gender, because this is analogous to the case where person A is deliberately trying to select an H coin to show us.

Of course, there's a continuum of outcomes here. For example, if we met the person at work in a profession where there's a different number of males vs. females.

That, at least, is how I understand this problem is resolved.

• Blake, you make good points. An insightful discussion on the boy-girl problem is in arxiv.org/pdf/1102.0173.pdf Commented Jun 10, 2015 at 9:10

The first answer is the correct one. The flaw in the reasoning in the second is that even when you are given that one of the coins came up H, it is twice as likely that the other came up T, this is fairly similar to the concept of permutations vs combinations, the TH possibility is a permutation because we can reorder it to HT, thus it represents $2$ seperate possibilities, while HH only represents one ordering, so it is a combination, and we have the correct probability of the other being tails, which is to say $\frac 2 3$

Person A has to explain the full rules before tossing the coins.

On the other hand, suppose person A instead shows person B that one of the coins came up heads (but does not say whether this is the first or second coin).

This is not enough information to calculate a probability.

Consider the following five situations, which are subtly different, and yet all correspond to your description:

Situation 1. Person A announces that if at least one of the coins is Heads, they will show a Heads coin to B, and if both coins are Tails, they will show a Tails coin. Person A tosses the two coins, then shows a coin to B, and this coin is Heads.

Situation 2. Person A announces that they will choose one of the two coins uniformly at random and show it to B. Person A tosses the two coins, then shows a coin to B, and this coin is Heads.

Situation 3. Person A decides that if they get Heads/Heads, they will show a Heads to B, but if they get at least a Tails, they will show a Tails to B. Person A tosses the two coins, then shows a coin to B, and this coin is Heads.

Situation 4. Person A, being mischievous, decides that if they get Heads/Tails, they'll show a Heads to B, but if they get Heads/Heads or Tails/Tails, then they won't show any coin. Person A tosses the two coins, then shows a coin to B, and this coin is Heads.

Situation 5. Person A doesn't announce the rules beforehand. Person A tosses the two coins, then shows a coin to B, and this coin is Heads.