ProbabilityHardBloom L3
Question
Ram has 4 bats namely $B_{1},B_{2},B_{3},B_{4}$. He randomly selects one of the three bats which was not used in previous match. If for his first match the chosen bat was $B_{1}$ and $P(n)$ denotes the probability that he chooses $B_{1}$ in $n^{\text{th}\text{~}}$ match, then
Options
A.$P(3) = \frac{1}{3}$
B.$P(5) = \frac{7}{27}$
C.$P(6) = \frac{10}{81}$
D.$P(7) = \frac{61}{243}$
Solution
Sol. [Key Concept: Recurrence $P(n) = \frac{1}{3}(1 - P(n - 1))$ for re-choosing the same bat.]
$$\begin{matrix} & P(n) = (1 - P(n - 1))\frac{1}{3} \\ & P(2) = 0 \\ & P(3) = \frac{1}{3},P(4) = \frac{2}{9},P(5) = \frac{7}{27} \\ & P(6) = \frac{20}{81},P(7) = \frac{61}{243} \end{matrix}$$
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