Question

Two dice are thrown together. Then, the probability that the sum of numbers appearing on them is a prime number, is
A. $\dfrac{5}{{12}}$
B. $\dfrac{7}{{18}}$
C. $\dfrac{{13}}{{36}}$
D. $\dfrac{{11}}{{36}}$

Answer 1

Hint: Probability is defined as the state of being probable and the extent to which something is likely to happen in the particular situations or the favourable outcomes.Probability of any given event is equal to the ratio of the favourable outcomes with the total number of the outcomes. Here we will first find the total possible outcomes and then the sum of the outcomes and then will identify the prime numbers.

Complete step by step answer:
Given that the two dice are thrown together.
Outcomes would be –
\[(1,1)\;{\text{(1,2) (1,3) (1,4) (1,5) (1,6)}} \\
\Rightarrow (2,1)\;{\text{(2,2) (2,3) (2,4) (2,5) (2,6)}} \\
\Rightarrow (3,1)\;{\text{(3,2) (3,3) (3,4) (3,5) (3,6)}} \\
\Rightarrow (4,1)\;{\text{(4,2) (4,3) (4,4) (4,5) (4,6)}} \\
\Rightarrow (5,1)\;{\text{(5,2) (5,3) (5,4) (5,5) (5,6)}} \\
\Rightarrow (6,1)\;{\text{(6,2) (6,3) (6,4) (6,5) (6,6)}} \\ \]
The total possible outcomes from the two thrown dices will be $ = 36$ ..... (A)
The favourable outcomes that are sum of the two dices should be prime numbers and the prime numbers between $2$ and $12$ are $2,3,5,7,11$
Favourable outcomes =
$(1,1){\text{ (1,2) (1,4) (1,6) (2,3) (2,5) (3,2) (3,4) (4,1) (4,3) (5,2) (5,6) (6,1)}}{\text{ (6,5)}}$
The total number of favourable outcomes $ = 14$ .... (B)
The probability that A wills –
$P(A) = \dfrac{{14}}{{36}}$
Find the factors for the above terms for the above fraction –
$P(A) = \dfrac{{2 \times 7}}{{2 \times 18}}$
Common factors from the numerator and the denominator in the above fraction cancel each other and therefore remove from the numerator and the denominator.
$\therefore P(A) = \dfrac{7}{{18}}$

Hence, option B is the correct answer.

Note:Be good at identifying the prime numbers, prime numbers are defined as the numbers which have only two factors : the number one and the number itself. Be good in multiples and division and always remove common factors from the numerator and the denominator to get the simplified form.