Question

Two dice are thrown simultaneously. Find the probability that the score obtained is a perfect square or a prime number.

Answer 1

Here, $S=\left\{\right(1,1).,(1,2),.......(6,5),(6,6\left)\right\}$
$\therefore$ number of all possible outcomes $n\left(s\right)=6\times 6=36$.
Let $E=$ event score obtained is a perfect square
and $F=$ event score obtained is a prime number
Then, $E=\left\{\right(1,3),(2,2),(3,6),(4,5),(5,4),(6,3\left)\right\}$
$\therefore$ $n\left(E\right)=6$
$F=\left[\right(1,1),(1,2),(1,4),(1,6),(2,1),(2,3),(2,5),(3,2),(3,4),(4,1),(4,3),(5,2),(5,6),(6,1),(6,5\left)\right]$
$\therefore$ $n\left(F\right)=15$
$P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}=\frac{6}{36}=\frac{1}{6}$
$P\left(F\right)=\frac{n\left(F\right)}{n\left(S\right)}=\frac{15}{36}=\frac{5}{12}$
The events $E$ and $F$ are mutually exclusive.
$\therefore P(E\cup F)=P\left(E\right)+P\left(F\right)=\frac{1}{6}+\frac{5}{12}=\frac{7}{12}$ [by addition theorem]