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

Question

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?

Answer 1

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 $$P=Pr\{(T,T),(HT,HT),(HHT,HHT),(HHHT,HHHT),...\}=\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{64}+...=\dfrac{1}{3}$$

Answer 2

The rounds that give two heads can be neglected.

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: $$P(T=2\mid T\geq1)=\frac{P(T=2)}{P(T\geq1)}=\frac{1/4}{3/4}=\frac13$$