Chapter 5: Probability and Random Variables

In each of Exercises 5.167-5.172, we have provided the number of trials and success probability for Bernoulli trials. LetX denote the total number of successes. Determine the required probabilities by using

(a) the binomial probability formula, Formula 5.4 on page 236. Round your probability answers to three decimal places.

(b) TableVII in AppendixA. Compare your answer here to that in part (a).

$n=5,p=\frac{3}{5},P(X=4)$

In each of Exercises 5.167-5.172, we have provided the number of trials and success probability for Bernoulli trials. LetX denote the total number of successes. Determine the required probabilities by using

(a) the binomial probability formula, Formula 5.4 on page 236. Round your probability answers to three decimal places.

(b) TableVII in AppendixA. Compare your answer here to that in part (a).

$n=4,p=\frac{1}{4},P(X=2)$

Pinworm Infestation. Use Procedure 5.1 on page 236 to solve part (g) of Exercise 5.165.

Psychiatric Disorders. Use Procedure 5.1 on page 236 to solve part (g) of Exercise 5.166.

Tossing a Coin. If we repeatedly toss a balanced coin, then, in the long run, it will come up heads about half the time. But what is the probability that such a coin will come up heads exactly half the time in 10 tosses?

Rolling a Die. If we repeatedly roll a balanced die, then, in the long run, it will come up "4" about one-sixth of the time. But what is the probability that such a die will come up "4" exactly once in six rolls?

According to the Daily Racing Farm, the probability is about \(0.67\) that the favorite in a horse race will finish in the money (first, second or third place). In the next five races, what is the probability that the favorite finishes in the money.

a. exactly twice?

b. exactly four times

c. at least four times?

d. between two and four times, inclusive?

e. Determine the probability distribution of the random variable \(X\), the number of times the favorite finishes in the money in the next five races.

f. Identify the probability distribution of \(X\) as right skewed, symmetric or left skewed without consulting its probability distribution or drawing its probability histogram.

g. Draw a probability histogram for \(X\).

h. Use your answer from part (c) and definitions \(5.9\) and \(5.10\) on pages \(227\) and \(229\) respectively to obtain the mean and standard deviation of the random variable \(X\).

i. Use formula \(5.5\) on page \(239\) to obtain the mean and standard deviation of the random variable \(X\).

j. Interpret your answer for the mean in words.

Horse Racing. According to the Daily Racing Form, the probability is about$0.67$that the favorite in a horse race will finish in the money (first, second, or third place). In the next five races, what is the probability that the favorite finishes in the money
a. exactly twice?
b. exactly four times?
c. at least four times?
d. between two and four times, inclusive?
e. Determine the probability distribution of the random variable$X$, the number of times the favorite finishes in the money in the next five races.
f. Identify the probability distribution of$X$ as right skewed, symmetric, or left skewed without consulting its probability distribution or drawing its probability histogram.
g. Draw a probability histogram for $X$h. Use your answer from part (e) and Definitions $5.9$ and $5.10$ on pages $227$and $229$, respectively, to obtain the mean and standard deviation of the random variable $X$
i. Use Formulawidth="24" height="20" role="math">5.5on page $239$ to obtain the mean and standard deviation of the random variable$X$.

The probability is \(0.314\) that the gestation period of a woman will exceed \(9\) months. In six human births, what is the probability that the number in which the gestation period exceeds \(9\) months is

a. exactly three?

b. exactly five?

c. at least five?

d. between three and five times, inclusive?

e. Determine the probability distribution of the random variable \(X\), the number of six human births in which the gestation period exceeds \(9\) months.

f. Identify the probability distribution of \(X\) as right skewed, symmetric or left skewed without consulting its probability distribution or drawing its probability histogram.

g. Draw a probability histogram for \(X\).

h. Use your answer from part (c) and definitions \(5.9\) and \(5.10\) on pages \(227\) and \(229\) respectively to obtain the mean and standard deviation of the random variable \(X\).

i. Use formula \(5.5\) on page \(239\) to obtain the mean and standard deviation of the random variable \(X\).

j. Interpret your answer for the mean in words.

Traffic Fatalities and Intoxication. The National Safety Council publishes information about automobile accidents in Accident Facts. According to that document, the probability is 0.40 that a traffic fatality will involve an intoxicated or alcohol-impaired driver or nonoccupant. In eight traffic fatalities, find the probability that the number,Y, that involve an intoxicated or alcohol-impaired driver or nonoccupant is

(a) exactly three; at least three; at most three.

(b) between two and four, inclusive.

(c) Find and interpret the mean of the random variable Y.

(d) Obtain the standard deviation of Y.