# Two dice are thrown together.

Question:

Two dice are thrown together. What is the probability that the sum of the numbers on the two faces is neither divisible by 3 nor by 4.

Solution:

Given: Two dice are thrown together.

Sample Space:

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

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

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

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

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

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

To Find: $\mathrm{P}$ (sum of faces neither divisible by 3 nor by 4 )

Sum $=\{2,3,4,5,6,7,8,9,10,11,12\}$

Sum neither divisible by 3 nor $4=\{2,5,7,10,11\}$

$P=\frac{\text { number of favourable outcomes }}{\text { total possible outcomes }}$

$P($ sum of faces neither divisible by 3 nor by 4$)=\frac{5}{11}$

Hence, probability is $\frac{5}{11}$.