# Coin chosen is two headed coin in this probability question

I have a probability question that reads:

Question:

A box has three coins. One has two heads, another two tails and the last is a fair coin. A coin is chosen at random, and comes up head. What is the probability that the coin chosen is a two headed coin.

My attempt:

P(two heads coin| given head) = P(two heads coin * given head)/P(given head)
= 1/3/2/3 = 1/2


Not sure whether this is correct?

Hint: No, it isn't. Let $H \equiv$ obtaining heads, $A \equiv$ picking a two-headed coin, $B \equiv$ picking a two-tailed coin, and $C \equiv$ picking a fair coin. Then observe that the probability of obtaining a head is: \begin{align*} P(H) &= P(A)\cdot P(H \mid A) + P(B)\cdot P(H \mid B) + P(C)\cdot P(H \mid C) \\ &= \dfrac{1}{3} \cdot \dfrac{2}{2} + \dfrac{1}{3}\cdot \dfrac{0}{2} + \dfrac{1}{3}\cdot \dfrac{1}{2} \end{align*}

• so my numerator is correct and my denominator is wrong. am i right? – lakesh Aug 9 '13 at 10:34
• No, the numerator is wrong as well. Using my notation, you have calculated $P(A)$ instead of $P(A \text{ and } H)$. – Adriano Aug 9 '13 at 10:35
• I don't think you're answering the question being asked. You correctly find $P(H)=1/2$, but the question was about $P(A\mid H)$ which your hint seems to say nothing about at all. – hmakholm left over Monica Aug 9 '13 at 12:17
• @HenningMakholm I was hoping that the OP knew the formula: $$P(A \mid H) = \dfrac{P(A) \cdot P(H \mid A)}{P(H)}$$ so I helped him out with the harder part, which was calculating the denominator. – Adriano Aug 9 '13 at 16:32

For such a small number of options its easy to count them

The possible outcomes are:

tails or tails using the double tail coin
heads or tails using the fair coin

All these outcomes are equally likely. How many of these are heads and of those how many use the double headed coin?

$$Answer = \frac{2}{3}$$

P(A∣H)=P(A) * P(H∣A) / P(H) --> Bayes Theorem

P(A) = 1/3, because 1 out of 3 coins are 2-headed

P(H|A) = 1, because given you have the 2-headed coin, then it is for sure (100%) you will get a head

P(H) = 1/2, because there are 3 heads and 3 tails in total, so it's 50% chance to get heads

Therefore, P(A|H) = (1/3) * 1 / (1/2) = 2/3

There are 3 heads in the box. The probability that you picked any one of those 3 is equal at 1/3 each. Two of the heads are on the same coin. So it’s 2/3 likely that’s the coin you picked.

The symmetry of this problem provides another way to look at it: you choose a coin at random and look at one of its faces at random. That face shows something (maybe heads, maybe tails).

First question:

What is the probability that the other face of the coin is the same as the face you can see?

The answer to that question is clearly $\frac23$, since two of the three coins have the same thing on both faces and the third coin has something different on the other face no matter which face you look at.

Second question:

In which case is the probability greater that the other face is the same as the face you can see: (A) you see heads; (B) you see tails; or are the probabilities the same?

By symmetry (because the conditions of the experiment are exactly the same even if we relabel every "heads" as "tails" and every "tails" as "heads"), neither probability is greater; they are the same.

So it doesn't matter which face you see when you first look at the coin; there is a $\frac23$ probability the other face is the same, even if the first face is heads (as it is in the original question).

• Indeed, the probability that the coin is two-headed, given that you know it has at least one head on it, is $\frac12.$ But that is not what the question asked. In the question as asked, you see only one side of the coin. It is possible then to draw the fair coin and not know that it has any heads face on it. – David K May 24 '15 at 19:10