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A manufacturer of Christmas tree light bulbs knows that 3% of its bulbs are defective. Find the probability that a string of 100 lights contains at most four defective bulbs using both the binomial and Poisson distributions.

We know that for a probability distribution function to be discrete, it must have two characteristics. One is that the sum of the probabilities is one. What is the other characteristic?

Use the following information to answer the next five exercises: Javier volunteers in community events each month. He does not do more than five events in a month. He attends exactly five events $35\%$of the time, four events $25\%$of the time, three events $20\%$of the time, two events $10\%$of the time, one event $5\%$of the time, and no events 5% of the time.

The average number of children a Japanese woman has in her lifetime is 1.37. Suppose that one Japanese woman is randomly chosen.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. Find the probability that she has no children.

e. Find the probability that she has fewer children than the Japanese average.

f. Find the probability that she has more children than the Japanese average.

The average number of children a Spanish woman has in her lifetime is 1.47. Suppose that one Spanish woman is randomly chosen.

a. In words, define the Random Variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. Find the probability that she has no children.

e. Find the probability that she has fewer children than the Spanish average.

f. Find the probability that she has more children than the Spanish average .

The chance of having an extra fortune in a fortune cookie is about 3%. Given a bag of 144 fortune cookies, we are interested in the number of cookies with an extra fortune. Two distributions may be used to solve this problem, but only use one distribution to solve the problem.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. How many cookies do we expect to have an extra fortune?

e. Find the probability that none of the cookies have an extra fortune.

f. Find the probability that more than three have an extra fortune.

g. As n increases, what happens involving the probabilities using the two distributions? Explain in complete sentences.

According to the South Carolina Department of Mental Health web site, for every 200 U.S. women, the average number who suffer from anorexia is one. Out of a randomly chosen group of 600 U.S. women determine the following.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution ofX. X ~ _____(_____,_____)

d. How many are expected to suffer from anorexia?

e. Find the probability that no one suffers from anorexia.

f. Find the probability that more than four suffer from anorexia.

The chance of an IRS audit for a tax return with over $\backslash (25,000$in income is about $2\%$per year. Suppose that $100$people

with tax returns over $\backslash )25,000$are randomly picked. We are interested in the number of people audited in one year. Use a

Poisson distribution to anwer the following questions.

a. In words, define the random variable$X.$

b. List the values that $X$may take on.

c. Give the distribution of$X.X~\_\_\_\_\_(\_\_\_\_\_,\_\_\_\_\_)$

d. How many are expected to be audited?

e. Find the probability that no one was audited.

f. Find the probability that at least three were audited.

Approximately $8\%$of students at a local high school participate in after-school sports all four years of high school. A group of $60$seniors is randomly chosen. Of interest is the number that participated in after-school sports all four years of

high school.

a. In words, define the random variable $X$.

b. List the values that $X$may take on.

c. Give the distribution of$X.X~\_\_\_\_\_(\_\_\_\_\_,\_\_\_\_\_)$

d. How many seniors are expected to have participated in after-school sports all four years of high school?

e. Based on numerical values, would you be surprised if none of the seniors participated in after-school sports all

four years of high school.

f. Based on numerical values, is it more likely that four or that five of the seniors participated in after-school sports

all four years of high school? Justify your answer numerically.

On average, Pierre, an amateur chef, drops three pieces of egg shell into every two cake batters he makes. Suppose

that you buy one of his cakes.

a. In words, define the random variable $X$.

b. List the values that $X$may take on.

c. Give the distribution of$X.X~\_\_\_\_\_(\_\_\_\_\_,\_\_\_\_\_)$

d. On average, how many pieces of egg shell do you expect to be in the cake?

e. What is the probability that there will not be any pieces of egg shell in the cake?

f. Let’s say that you buy one of Pierre’s cakes each week for six weeks. What is the probability that there will not

be any egg shell in any of the cakes?

g. Based upon the average given for Pierre, is it possible for there to be seven pieces of shell in the cake? Why?

In words, the random variable$X=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

a. the number of times Mrs. Plum’s cats wake her up each week.

b. the number of times Mrs. Plum’s cats wake her up each hour.

c. the number of times Mrs. Plum’s cats wake her up each night.

d. the number of times Mrs. Plum’s cats wake her up.