Q.97 In major league baseball, a no-h... [FREE SOLUTION]

91影视

Introductory Statistics

Barbara Illowsky, Susan Dean

$Math Studyset 91影视 Explanations$ Math

OER 2018 Edition 路 902 pages

ISBN: 9781938168208

Chapter 5: Q.97 (page 357)

In major league baseball, a no-hitter is a game in which a pitcher, or pitchers, doesn't give up any hits throughout the game. No-hitters occur at a rate of about three per season. Assume that the duration of time between no-hitters is exponential.
a. What is the probability that an entire season elapses with a single no-hitter?
b. If an entire season elapses without any no-hitters, what is the probability that there are no no-hitters in the following season?
c. What is the probability that there are more than $3$ no-hitters in a single season?

Short Answer

Expert verified

a). The probability that an entire season elapses with a single no-hitter is $0.0498$ .

b). The value of $P (T > 2 鈭� T > 1)$ is $0.0498$ .

c). The value of $P (x > 3)$ is $0.3528$ .

Step by step solution

Part (a) Step 1: Given Information

$X ~ Exp (3)$

Where.

$m = 3$

$渭 = \frac{1}{3}$

$= 0.3333$

Part (a) Step 2: Explanation

The required probability can be calculated as below:

$P (x > 1) = 1 - P (x < 1)$

$= 1 - (1 - e^{- 3 - 1})$

= e^{- 3}

$= 0.0498$

Part (b) Step 1: Given Information

$X ~ Exp (3)$

Where.

$m = 3$

$渭 = \frac{1}{3}$

$= 0.3333$

Part (b) Step 2: Explanation

The required probability can be written as $P (T > 2 鈭� T > 1)$ . Now, use the memory less property of exponential distribution. That is,

$P (x > t + r 鈭� x > t) = P (x > r)$

So,

$P (T > 2 鈭� T > 1) = P (T > 1 + 1 鈭� T > 1)$

$= P (T > 1)$

$= 0.0498$

Therefore, the value of $P (T > 2 鈭� T > 1)$ is $0.0498$ .

Part (c) Step 1: Given Information

$X ~ Exp (3)$

Where.

$m = 3$

$渭 = \frac{1}{3}$

$= 0.3333$

Part (c) Step 2: Explanation (Part c)

Let $X =$ the number of no-hitters is a season. Assume that $X$ is Poisson with mean $位 = 3$ . Then,

localid="1650470480426" $P (X > 3) = 1 - P (X 鈮� 3) = 0.3528$ .

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91影视

Short Answer

Step by step solution

Part (a) Step 1: Given Information

Part (a) Step 2: Explanation

Part (b) Step 1: Given Information

Part (b) Step 2: Explanation

Part (c) Step 1: Given Information

Part (c) Step 2: Explanation (Part c)

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