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.
To combine the results we simply add them.
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
$$\Bbb P(S)=\Bbb P(S\mid E)\Bbb P(E)+\Bbb P(S\mid U)\Bbb P(U)\;.$$