ProbabilityHardBloom L3
Question
A, B, C are alternately and, in this order, rolling a fair die. The first one to roll an even number wins and the game is ended, then
Options
A.$P(A) = \frac{4}{7}$
B.$P(C) = \frac{1}{7}$
C.$P(B)P(C) = \frac{1}{14}$
D.$P(B) = \frac{3}{7}$
Solution
Sol. [Key Concept: Geometric series for the first success; $P(A) + P(B) + P(C) = 1 \Rightarrow \ P = \frac{4}{7}$.]
Let
$$\begin{matrix} & P(\text{ }A) = P \\ & P(\text{ }B) = \frac{1}{2}P \\ & P(C) = \frac{1}{2} \times \frac{1}{2} \times P \end{matrix}$$
$$\begin{matrix} & & \ \ \ \ \ \ \ \ & P(\text{ }A) + P(\text{ }B) + P(C) & \ = 1 \\ \Rightarrow & & \ \ \ \ \ \ \ \ & \frac{7}{4}P & \ = 1 \Rightarrow P = \frac{4}{7} \end{matrix}$$
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