Question

In a simultaneous throw of a pair of dice, if the probability of getting a sum less then $6$ is $\frac{5}{a}.$ Find $a.$

Answer 1

Let $S=$ Total outcomes

$E=$ Favourable outcomes (sum less than $6$)

$S=\left\{\right(1,1\left)\right(1,2\left)\right(1,3\left)\right(1,4\left)\right(1,5\left)\right(1,6)$

$(2,1)(2,2)(2,3)(2,4)(2,5)(2,6)$

$(3,1)(3,2)(3,3)(3,4)(3,5)(3,6)$

$(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)$

$(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)$

$(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)\}$

$n\left(S\right)=36$

$E=\left\{\right(1,1\left)\right(1,2\left)\right(1,3\left)\right(1,4\left)\right(2,1\left)\right(2,2\left)\right(2,3\left)\right(3,1\left)\right(3,2\left)\right(4,1\left)\right\}$

$n\left(E\right)=5$

$\Rightarrow P\left(E\right)=\frac{n\left(E\right)}{n\left(S\right)}=\frac{10}{36}=\frac{5}{18}$

i.e probability of getting a sum less than $6$ is $\frac{5}{18}=\frac{5}{a}$

$\therefore a=18$