# What is the probability the two-headed coin is chosen from among the coins that came up heads?

An urn contains 3 coins; 2 biased with $P(H) = p$, and the other is a two-headed coin. All the coins are tossed at once. If a coin is selected from those that came up heads, what is the probability that it was the two-headed coin?

-
Since you are new, I want to give some advice about the site: To get the best possible answers, you should explain what your thoughts on the problem are so far. That way, people won't tell you things you already know, and they can write answers at an appropriate level; also, people tend to be more willing to help you if you show that you've tried the problem yourself. If this is homework, please add the [homework] tag; people will still help, so don't worry. – Zev Chonoles Apr 8 '13 at 0:07

## 2 Answers

Hint: For each $n\in\{0,1,2,3\}$,

• calculate the probability $P_n$ that there are exactly $n$ heads.

• calculate the probability $Q_n$ that, if $n$ heads have appeared, then the two-headed coin is chosen

Then the probability that the two-headed coin is chosen overall is $$P_0Q_0+P_1Q_1+P_2Q_2+P_3Q_3.$$ (Do you understand why?)

-
very helpful thank you – user71501 Apr 8 '13 at 2:22

HINT: The possible results are three heads, with probability $p^2$; two heads and a tail, with probability $2p(1-p)$; and one head and two tails, with probability $(1-p)^2$. That gives half of the information embodied in the following table:

$$\begin{array}{rccc} &3\text{ heads}&2\text{ heads}&1\text{ head}\\ \text{Probability}:&p^2&2p(1-p)&(1-p)^2\\ \text{Probability of choosing a head}:&?&?&? \end{array}$$

You should have no trouble completing the table. How can you use these six numbers to calculate the desired probability?

-