First slide

Theorems of probability

Question

A fair die is tossed repeatedly. A wins if it is 1 or 2 on two consecutive tosses and B wins if it is 3, 4, 5 or 6 on two consecutive tosses. The probability that A wins if the die is tossed indefinitely is

Moderate

A

1/3
B

5/21
C

1/4
D

2/5

Solution

Let
$\begin{array}{l} P (S) = P (1 or 2) = 1 / 3 \\ P (F) = P (3 or 4 or 5 or 6) = 2 / 3 \\ P (A wins) = P [(S S or S F S S or S F S F S S or \dots) \end{array}$
or $(FSS or FSFSS or \dots)]$
$\begin{matrix} = \frac{\frac{1}{9}}{1 - \frac{2}{9}} + \frac{\frac{2}{27}}{1 - \frac{2}{9}} \\ = \frac{1}{9} \times \frac{9}{7} + \frac{2}{27} \times \frac{9}{7} \\ = \frac{1}{7} + \frac{2}{21} = \frac{3 + 2}{21} = \frac{5}{21} \end{matrix}$
$P (A winning) = \frac{5}{21}, P (B winning) = \frac{16}{21}$

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