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Q. 5.1

Expert-verified

A First Course in Probability

Found in: Page 217

Book edition 9th

Author(s) Sheldon M. Ross

Pages 432 pages

ISBN 9780321794772

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Short Answer

The number of minutes of playing time of a certain high school basketball player in a randomly chosen game is a random variable whose probability density function is given in the following figure:

Find the probability that the player plays
(a) more than $15$ minutes;
(b) between $20 a n d 35$ minutes;
(c) less than $30$ minutes;
(d) more than $36$ minutes

(a) The probability that the player plays more than $15$ minutes is $0.875$

(b) The probability that the player plays between $20 a n d 35$ is $0.625$

(c) The probability that the player plays less than $30$ minutes is $0.75$

(d) The probability that the player plays more than $36$ minutes is $0.1$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: Find the probability that the player plays more than 15 minutes (part a)

Formalize the given probability function. As may be observed from the graph,

$f (x) = 0.025, x \in [10, 20) \cup (30, 40]$

$f (x) = 0.05, x \in [20, 30]$

$f (x) = 0$ , otherwise

$X$ is the random variable with the density function defined $P$ . The needed probabilities are calculated as the integrals of the density function $f$ over the relevant intervals.

$P (X > 15) = 1 - P (X \leq 15) = 1 - \int_{10}^{15} f (x) d x = 1 - \int_{10}^{15} 0.025 d x$

$= 1 - 0.025 \times 5 = 0.875$

Step2: Find the probability that the player plays between 20 and 35 minutes (part b)

$P (X \in (20, 35)) = P (X \in (20, 30)) + P (X \in (30, 35))$

$= \int_{20}^{30} f (x) d x + \int_{30}^{35} f (x) d x$

$= 0.05 \times 10 + 0.025 \times 5 = 0.625$

Step3: Find the probability that the player plays less than 30 minutes (part c)

$P (X < 30) = 1 - P (X \geq 30) = 1 - \int_{30}^{40} f (x) d x = 1 - \int_{30}^{40} 0.025 d x$

$= 1 - 0.025 \times 10 = 0.75$

step4: Find the probability that the player plays more than 36 minutes (part d)

$P (X > 36) = \int_{36}^{40} f (x) d x = \int_{36}^{40} 0.025 d x = 0.025 \times 4 = 0.1$

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