# Probability that both people flip their first tails on the same round?

Sarah and Paula are flipping coins. During each round, each one flips a fair coin and then they compare their results. What is the probability that both girls flip their first tails during the same round?

• And what have you done so far on this? – Macavity Mar 18 '18 at 15:13
• This is equivalent to asking, on a given series of flips, what is the probability of getting an even number of heads followed by two tails. I'm not sure if this is an easier situation to analyze, but it does have the benefit of only dealing with one series of coin flips instead of two. – Dennis Muhonen Mar 18 '18 at 15:40

Since they flip the coins independently the event is in fact which each of them draws $\{T,HT,HHT,HHHT,...\}$ with probability so the general probability is $$P=Pr\{(T,T),(HT,HT),(HHT,HHT),(HHHT,HHHT),...\}=\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{64}+...=\dfrac{1}{3}$$
Looking at the first round that gives at least one tail let $T$ denotes the number of tails is that round.
Then you are asked to find: $$P(T=2\mid T\geq1)=\frac{P(T=2)}{P(T\geq1)}=\frac{1/4}{3/4}=\frac13$$