Question

A pair of fair dice is tossed one time. What is the probability that ⦁ The sum of two faces is a prime number? ⦁ The sum of two faces is more than 9? ⦁ The sum of two faces is between 5 and 9? ⦁ The sum of two faces is at most 6? ⦁ The sum of two faces is at least 8?

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c} & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hdashline 2 & 3 & 4 & 5 & 6 & 7 & 8\\ \hdashline 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hdashline 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hdashline 5 & 6 & 7 & 8 & 9 & 10 & 11 \\ \hdashline 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hdashline \end{array}$$

1.

Prime numbers: $2,3,5,7,11$

$$P(sum\ is\ prime)=\dfrac{1+2+4+6+2}{36}=\dfrac{5}{12}$$

2.

$$P(sum>9)=\dfrac{3+2+1}{36}=\dfrac{1}{6}$$

3.

$$P(5\leq sum\leq 9)=\dfrac{4+5+6+5+4}{36}=\dfrac{2}{3}$$

$$P(5< sum<9)=\dfrac{5+6+5}{36}=\dfrac{4}{9}$$

4.

$$P( sum\leq 6)=\dfrac{1+2+3+4+5}{36}=\dfrac{5}{12}$$

5.

$$P(sum\geq 8)=\dfrac{5+4+3+2+1}{36}=\dfrac{5}{12}$$