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Q.

Sachin has ‘4’ bats namely B₁, B₂, B₃, B₄. He randomly selects one out of the three bats which was not used in previous match. If for his first match the chosen bat was B₁ and P(n) denotes the probability that he chooses bat B₁in n^th match, then which of the following is/are true?

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a

$P (4) = \frac{2}{9}$

b

$P (7) = \frac{61}{243}$

c

$P (3) = \frac{1}{3}$

d

P(2) = 0

answer is A, B, C, D.

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Detailed Solution

$\begin{array}{l} P (1) = 1; P (2) = 0 and 1 - P (n - 1) \\ \Rightarrow P (n + 1) = \frac{1}{3} (1 - P (n)) ⟶ (2) \\ (1) - (2) \Rightarrow P (n + 1) - P (n) = \frac{1}{3} (P (n - 1) - P (n)) \end{array}$
Let $Q (n) = P (n + 1) - P (n) ⟶ (3)$
$\begin{array}{l} Q (n) = \frac{- 1}{3} Q (n - 1) \\ \Rightarrow Q (n) is G.P with 1^{st} term Q (1) = - 1 and common \\ ratio \frac{- 1}{3} \\ \Rightarrow \sum_{i = 1}^{n} Q (i) + P (1) {Summation on (3) \end{array}$
$\Rightarrow p (n) = \sum_{i = 1}^{n - 1} Q (i) + P (1) = - 1 \{\frac{{(\frac{- 1}{3})}^{n - 1} - 1}{\frac{- 4}{3}}\} + 1$

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Sachin has ‘4’ bats namely B1, B2, B3, B4. He randomly selects one out of the three bats which was not used in previous match. If for his first match the chosen bat was B1 and P(n) denotes the probability that he chooses bat B1in nth match, then which of the following is/are true?