A man alternatively tosses a fair coin and rolls a fair ordinary dice. He starts with the coin. The probability that he gets a tail on the coin before getting 5 or 6 on the dice, is equal to

$\frac{3}{4}$

Probability of getting 5 or 6 on a specific roll of dice

$\frac{2}{6}=\frac{1}{3}$

and, probability of getting a tail = $\frac{1}{2}$

The desired outcome can happen, in general, on (2r + 1)^th trial. That means first 2r trials should result neither in tail nor in 5 or 6, and (2r + 1)^th trial must result in tail.

It the conesponding probability is p_r then

$p_r=\left(\frac{1}{2}\right)^r . \left(\frac{2}{3}\right)^r . \frac{1}{2}=\frac{1}{2} . \left(\frac{1}{3}\right)^r$

Thus, required probability

$=\sum\limits_{r=0}^{\infty} p_r=\frac{1}{2} \sum\limits_{r=0}^{\infty}\left(\frac{1}{3}\right)^r$

$=\frac{3}{4}$