First slide

Theorems of probability

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

Moderate

A

1/2
B

7/18
C

3/4
D

5/12

Solution

Consider two events as follows:
A_i: getting number i on first die
B_i: getting a number more than i on second die
The required probability is
$\begin{array}{l} P (A_{1} \cap B_{1}) + P (A_{2} \cap B_{2}) + P (A_{3} \cap B_{3}) + P (A_{4} \cap B_{4}) \\ + P (A_{5} \cap B_{5}) = \sum_{i = 1}^{5} P (A_{i} \cap B_{i}) = \sum_{i = 1}^{5} P (A_{i}) P (B_{i}) \\ [∵ A_{i}, B_{i} are independent] \\ = \frac{1}{6} [P (B_{1}) + P (B_{2}) + \dots + P (B_{5})] \\ = \frac{1}{6} (\frac{5}{6} + \frac{4}{6} + \frac{3}{6} + \frac{2}{6} + \frac{1}{6}) = \frac{5}{12} \end{array}$

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