Conditional probability of coin tosses

Question

Suppose we have the following events:

• X = "Number of heads in the first two tosses"
• Z = "Number of tosses before first head".

I have to compute $P(Z=1 | X=1)$. I can proceed as follows: $$ P(Z=1 | X=1) = \frac{P(Z=1 , X=1)}{P(X=1)}.$$

I can compute easily with the Binomial the denominator but I have no idea about the numerator. I know that the result should be $pq$ if the probability of getting head is $p$. Is there a way to visualize this probability?

Answer 1

If the number of tosses before the first head is $1$, then in the first two tosses you have necessarily gotten exactly a single head, which means that $$ (Z = 1)\subseteq (X = 1) $$ as events. Consequently $$ P(Z = 1, X = 1) = P(Z = 1) $$