First slide

Theorems of probability

Question

A pair of unbiased dice is rolled together till a sum of either 5 or 7 is obtained. The probability that 5 comes before 7 is

Moderate

A

2/5
B

3/5
C

4/5
D

None of these

Solution

Let A denote the event that a sum of 5 occurs, B the event that a sum of 7 occurs, and C the event that neither a sum
of 5 nor a sum of 7 occurs.
We have,
$\begin{aligned} P (A) = \frac{4}{36} = \frac{1}{9}, P (B) = \frac{6}{36} = \frac{1}{6} \\ P (C) = \frac{26}{36} = \frac{13}{18} \end{aligned}$
Thus, probability that A occurs before B is
$P [A or (C \cap A) or (C \cap C \cap A) or \dots] = P (A) + P (C \cap A) + P (C \cap C \cap A) + \dots$
$\begin{array}{l} = P (A) + P (C) P (A) + P (C)^{2} P (A) + \dots \\ = \frac{1}{9} + (\frac{13}{18}) \times \frac{1}{9} + {(\frac{13}{18})}^{2} \frac{1}{9} + \dots \\ = \frac{1 / 9}{1 - 13 / 18} = \frac{2}{5} \end{array}$

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