# Flipping a coin 2 times

We have two coins one fair and the other biased for which the probability of head is 2/3. We choose a coin at random, and we flip it two times. What is the probability that in both flips we will receive identical results? Obviously i know how to find single events probabilities however i do not know how to combine everything to get a desired result.

• PS: the terms are "fair" and "biased" not "correct" or "incorrect". – Graham Kemp Oct 29 '15 at 19:59
• my mistake, i am not a native speaker :) – mkropkowski Oct 30 '15 at 12:45

${1\over2}({1\over4}+{1\over4})+{1\over2}({4\over9}+{1\over9})={19\over36}$
• @adhg The probability we choose the biased coin is $1\over 2$, the probability that it ends up with two heads is $({2\over 3})^2$ and similar for the two tails case $({1\over 3})^2$. – cr001 Nov 3 '17 at 2:15
HINT: Let $S$ be the event of getting identical results, $F$ the event of choosing the fair coin, and $U$ the event of picking the unfair coin. Then
$$\Bbb P(S)=\Bbb P(S\mid E)\Bbb P(E)+\Bbb P(S\mid U)\Bbb P(U)\;.$$