Probability two sequences of coin flips reach consecutive heads at the same time.

Question

Two people start flipping coins. The probability of heads is 0.5 (bonus if you can do it for $p_1$ and $p_2$). What is the probability that both will hit two consecutive heads simultaneously (as opposed to one of them doing so before the other)?

Answer 1

Ok, figured it out. Thought I'd post the answer instead of deleting (BTW, I really don't understand the downvote - especially without having the courtesy to add a comment).

Let A be the event the two sequences reach HH simultaneously. Let's condition on the result of the first toss for each of them.

$P(A) = \frac{P(A|HH)+2*P(A|HT)+P(A|TT)}{4}$ Also,

$P(A|TT)=P(A)$

since it just resets.

$P(A|HH)=1/4+P(A)/4$

(If we get another HH, we're done. If we get TT, we reset. All other cases, it doesn't happen).

$P(A|HT) = 1/2(P(A|HT)/2+P(A)/2)$

Solving these, we get $P(A) = 3/47$