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

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)?

• I don't see why the probability isn't 1. If both are flipping coins forever, at some time it will happen that both get heads. Are there other rules to the flipping? – Jasper Sep 26 '18 at 6:03
• Sorry, I should have mentioned, this is before any one of them flips two heads before the other. – Rohit Pandey Sep 26 '18 at 6:10
• Sorry, it was partly my own mistake, I missed "consecutive" in the title. – Jasper Sep 26 '18 at 16:28

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 $$P(A) = 3/47$$

• The downvote wasn't me, but it may have been for not mentioning what efforts you'd made to solve the problem. – Geoffrey Brent Sep 26 '18 at 5:43