Question

Two dices are rolled one after the other. The probability that the number on the first is smaller than the number on the second is

• A. $\frac{1}{2}$
• B. $\frac{7}{18}$
• C. $\frac{3}{4}$
• D. $\frac{5}{12}$

Answer 1

The correct option is D $\frac{5}{12}$

Consider the two events as follows:
$A_i:$ getting number $i$ on first die
$B_i:$ getting a number more than $i$ on second die
The recquired probability is
$P(A_1\cap B_1)+P(A_2\cap B_2)+P(A_3\cap B_3)P(A_4\cap B_4)+P(A_5\cap B_5)=\sum _{i=1}^{5}P(A_i\cap B_i)=\sum _{i=1}^{5}P\left(A_i\right)P\left(B_i\right)$
$[\because A_i,B_i\text{are independent events}]$
$=\frac{1}{6}\left[P\right(B_1)+P(B_2)+\cdots +P(B_5\left)\right]$
$=\frac{1}{6}(\frac{5}{6}+\frac{4}{6}+\frac{3}{6}+\frac{2}{6}+\frac{1}{6})=\frac{5}{12}$