SATHEE: Probability 5 Question 16

Probability 5 Question 16

16. Suppose the probability for $A$ to win a game against $B$ is 0.4 . If $A$ has an option of playing either a best of 3 games’ or a ‘best of 5 games" match against $B$ , which option should choose so that the probability of his winning the match is higher? (no game ends in a draw).

$(1989, 5$ M)

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Answer:

Correct Answer: 16. Best of 3 games

Solution:

Case I When $A$ plays 3 games against $B$ .

In this case, we have $n = 3, p = 0.4$ and $q = 0.6$

Let $X$ denote the number of wins. Then,

$P (X = r) =^{3} C_{r} (0.4)^{r} (0.6)^{3 - r}; r = 0, 1, 2, 3$

$∴ P_{1} =$ probability of winning the best of 3 games

$\begin{aligned} = P (X \geq 2) \\ = P (X = 2) + P (X = 3) \\ =^{3} C_{2} (0.4)^{2} (0.6)^{1} +^{3} C_{3} (0.4)^{3} (0.6)^{0} \\ = 0.288 + 0.064 = 0.352 \end{aligned}$

Case II When $A$ plays 5 games against $B$ .

In this case, we have

$n = 5, p = 0.4 and q = 0.6$

Let $X$ denotes the number of wins in 5 games.

Then,

$P (X = r) =^{5} C_{r} (0.4)^{r} (0.6)^{5} - r$ , where $r = 0, 1, 2 \dots, 5$

$∴ P_{2} =$ probability of winning the best of 5 games

$= P (X \geq 3)$

$= P (X = 3) + P (X = 4) + P (X = 5)$

$=^{5} C_{3} (0.4)^{3} (0.6)^{2} +^{5} C_{4} (0.4)^{4} (0.6) +^{5} C_{5} (0.4)^{5} (0.6)^{0}$

$= 0.2304 + 0.0768 + 0.1024 = 0.31744$ Clearly, $P_{1} > P_{2}$ . Therefore, first option i.e. ‘best of 3 games’ has higher probability of winning the match.

Probability 5 Question 16

16. Suppose the probability for $A$ to win a game against $B$ is 0.4 . If $A$ has an option of playing either a best of 3 games’ or a ‘best of 5 games" match against $B$ , which option should choose so that the probability of his winning the match is higher? (no game ends in a draw).

Answer:

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Probability 5 Question 16

16. Suppose the probability for A to win a game against B is 0.4 . If A has an option of playing either a best of 3 games’ or a ‘best of 5 games" match against B, which option should choose so that the probability of his winning the match is higher? (no game ends in a draw).

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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16. Suppose the probability for $A$ to win a game against $B$ is 0.4 . If $A$ has an option of playing either a best of 3 games’ or a ‘best of 5 games" match against $B$ , which option should choose so that the probability of his winning the match is higher? (no game ends in a draw).