Question

A pair of dice is thrown $4$ times. If getting a doublet is considered a success, find the probability of two successes.

Answer 1

Consider $S$ as the event of success and $F$ as the event of failure

Given that getting a doublet is considered as a success

The order should contain two failures and two success events which can occur in any pattern

The number of patterns are arranging $2S^{\prime }s$ and $2F^{\prime }s$ in $4$ blanks,

$\therefore$ The total number of ways are $\frac{4!}{2!}=24$

Now, the number of chances of failing are $6.5=30$

because the number in the first die is not same as the second one, as there are $6$ cases for the first die,

$\therefore$ There will be $5$ chances for the second one,

Now, the number of chances for succeeding are

$6$ ways, because both the numbers in the dice are same

$\therefore$ There are $6$ chances in total.

$\therefore$ The total favourable events for getting two doublets are

$\left(\mathrm{6.6.30.30}\right).6=194400$,

The total cases are $\mathrm{36.36.36.36}=1679616$

$\therefore$ the probability of getting a doublet is $\frac{194400}{1679616}=0.1157$