Chapter 5: Q. 5.1 (page 217)

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

Short Answer

Expert verified

(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 $0.625$

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

Step by step solution

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.

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

$= 1 - 0.025 \times 5 = 0.875$

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

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

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

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

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

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

$= 1 - 0.025 \times 10 = 0.75$

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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Short Answer

Step by step solution

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

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

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

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

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