5. Binomial Distribution & Discrete Random Variables

In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.

Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.

a. Find the mean number of births per day.

In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.

Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.

b. Find the probability that in a single day, there are 16 births.

In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.

Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.

c. Find the probability that in a single day, there are no births. Would 0 births in a single day be a significantly low number of births?

Random Variable The accompanying table lists probabilities for the corresponding numbers of unlicensed software packages when four software packages are randomly selected in China. What is the random variable, what are its possible values, and are its values numerical?

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Finding the Mean, Variance, and Standard Deviation In Exercises 29–34, (a) find the mean, variance, and standard deviation of the probability distribution, and (b) interpret the results.

Dogs The number of dogs per household in a neighborhood

Finding the Mean, Variance, and Standard Deviation In Exercises 29–34, (a) find the mean, variance, and standard deviation of the probability distribution, and (b) interpret the results.

Machine Parts The number of defects per 1000 machine parts inspected

Finding an Expected Value In Exercises 37 and 38, find the expected value E(x) to the player for one play of the game. If x is the gain to a player in a game of chance, then E(x) is usually negative. This value gives the average amount per game the player can expect to lose.

In American roulette, the wheel has the 38 numbers, 00, 0, 1, 2, . . ., 34, 35, and 36, marked on equally spaced slots. If a player bets $1 on a number and wins, then the player keeps the dollar and receives an additional $35. Otherwise, the dollar is lost.

Finding an Expected Value In Exercises 37 and 38, find the expected value E(x) to the player for one play of the game. If x is the gain to a player in a game of chance, then E(x) is usually negative. This value gives the average amount per game the player can expect to lose.

A high school basketball team is selling $10 raffle tickets as part of a fund-raising program. The first prize is a trip to the Bahamas valued at $5460, and the second prize is a weekend ski package valued at $496. The remaining 18 prizes are $100 gas cards. The number of tickets sold is 3500.

In Exercises 1–4, find the indicated probability using the geometric distribution.

Find P(3) when p = 0.65

In Exercises 1–4, find the indicated probability using the geometric distribution.

Find P(5) when p = 0.09

In Exercises 5–8, find the indicated probability using the Poisson distribution.

P(3) when μ = 6

Geometric Distribution: Mean and Variance In Exercises 29 and 30, use the fact that the mean of a geometric distribution is μ = 1/p and the variance is

sigma^2 = q/p^2

Paycheck Errors A company assumes that 0.5% of the paychecks for a year were calculated incorrectly. The company has 200 employees and examines the payroll records from one month. (a) Find the mean, variance, and standard deviation. (b) How many employee payroll records would you expect to examine before finding one with an error?

In Exercises 1 and 2, determine whether the random variable x is discrete or continuous. Explain.

Let x represent the grade on an exam worth a total of 100 points.

In your own words, describe the difference between the value of x in a binomial distribution and in the Poisson distribution.

Using a Distribution to Find Probabilities In Exercises 11–26, find the indicated probabilities using the geometric distribution, the Poisson distribution, or the binomial distribution. Then determine whether the events are unusual. If convenient, use a table or technology to find the probabilities.

Immigration The mean number of people who immigrated to the United States per hour was about 5.5 in April 2021. Find the probability that the number of people who immigrate to the U.S. in a given hour in April 2021 was (a) zero, (b) exactly five, and (c) exactly eight. (Source: U.S. Census Bureau)