4.6 Poisson Distribution (Optional)

The average number of fish caught in an hour is eight. Of interest is the number of fish caught in 15 minutes. The time interval of interest is 15 minutes. What is the average number of fish caught in 15 minutes?

A bank expects to receive six bad checks per day, on average. What is the probability of the bank getting fewer than five bad checks on any given day? Of interest is the number of checks the bank receives in one day, so the time interval of interest is one day. Let X = the number of bad checks the bank receives in one day. If the bank expects to receive six bad checks per day then the average is s…

An electronics store expects to have 10 returns per day on average. The manager wants to know the probability of the store getting fewer than eight returns on any given day. State the probability question mathematically.

This is a Poisson problem because you are interested in knowing the number of times the news reporter says "uh" during a broadcast.

a. What is the interval of interest?

This is a Poisson problem because you are interested in knowing the number of times the news reporter says "uh" during a broadcast.

b. What is the average number of times the news reporter says "uh" during one broadcast?

This is a Poisson problem because you are interested in knowing the number of times the news reporter says "uh" during a broadcast.

c. Let X = ________. What values does X take on?

This is a Poisson problem because you are interested in knowing the number of times the news reporter says "uh" during a broadcast.

d. The probability question is P(______).

An emergency room at a particular hospital gets an average of five patients per hour. A doctor wants to know the probability that the ER gets more than five patients per hour. Give the reason why this would be a Poisson distribution.

A customer service center receives about 10 emails every half-hour. What is the probability that the customer service center receives more than four emails in the next six minutes? Use the TI-83+ or TI-84 calculator to find the answer.

According to Baydin, an email management company, an email user gets, on average, 147 emails per day. Let X = the number of emails an email user receives per day. The discrete random variable X takes on the values x = 0, 1, 2 . . . . The random variable X has a Poisson distribution: X ~ P(147). The mean is 147 emails.

1. What is the probability that an email user receives exactly 160 emails per day?
…

According to a recent poll girls between the ages of 14 and 17 send an average of 187 text messages each day. Let X = the number of texts that a girl aged 14 to 17 sends per day. The discrete random variable X takes on the values x = 0, 1, 2 …. The random variable X has a Poisson distribution: X ~ P(187). The mean is 187 text messages.

1. What is the probability that a teen girl sends exactly 175 tex…

Text message users receive or send an average of 41.5 text messages per day.

1. How many text messages does a text message user receive or send per hour?
2. What is the probability that a text message user receives or sends two messages per hour?
3. What is the probability that a text message user receives or sends more than two messages per hour?

Scientists recently researched the busiest airport in the world. On average, there are 2,500 arrivals and departures each day.

1. How many airplanes arrive and depart the airport per hour?
2. What is the probability that there are exactly 100 arrivals and departures in one hour?
3. What is the probability that there are at most 100 arrivals and departures in one hour?

On May 13, 2013, starting at 4:30 p.m., the probability of low seismic activity for the next 48 hours in Alaska was reported as about 1.02 percent. Use this information for the next 200 days to find the probability that there will be low seismic activity in 10 of the next 200 days. Use both the binomial and Poisson distributions to calculate the probabilities. Are they close?

On May 13, 2013, starting at 4:30 p.m., the probability of moderate seismic activity for the next 48 hours in the Kuril Islands off the coast of Japan was reported at about 1.43 percent. Use this information for the next 100 days to find the probability that there will be low seismic activity in 5 of the next 100 days. Use both the binomial and Poisson distributions to calculate the probabilities.…

Use the following information to answer the next six exercises: On average, a clothing store gets 120 customers per day.

What values does X take on?

Use the following information to answer the next six exercises: On average, a clothing store gets 120 customers per day.

What is the probability of getting 35 customers in the first four hours? Assume the store is open 12 hours each day.

Use the following information to answer the next six exercises: On average, a clothing store gets 120 customers per day.

What is the probability that the store will have fewer than 12 customers in the first two hours?

Use the following information to answer the next six exercises: On average, eight teens in the United States die from motor vehicle injuries per day. As a result, states across the country are debating raising the driving age.

Assume the event occurs independently in any given day. In words, define the random variable X.

Use the following information to answer the next six exercises: On average, eight teens in the United States die from motor vehicle injuries per day. As a result, states across the country are debating raising the driving age.

What values does X take on?

Use the following information to answer the next six exercises: On average, eight teens in the United States die from motor vehicle injuries per day. As a result, states across the country are debating raising the driving age.

Is it likely that there will be no teens killed from motor vehicle injuries on any given day in the United States? Justify your answer numerically.

The switchboard in a Minneapolis law office gets an average of 5.5 incoming phone calls during the noon hour on Mondays. Experience shows that the existing staff can handle up to six calls in an hour. Let X = the number of calls received at noon.

1. Find the mean and standard deviation of X.
2. What is the probability that the office receives at most six calls at noon on Monday?
3. Find the probability tha…

A manufacturer of decorative string lights knows that 3 percent of its bulbs are defective. Using both the binomial and Poisson distributions, find the probability that a string of 100 lights contains at most four defective bulbs.

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

1. In words, define the random variable X.
2. List the values that X may take on.
3. Give the distribution of X. X ~ _____(_____,_____)
4. Find the probability that she has no children.
5. Find the probability that she has fewer children than the Spanish average.
6. Find the probability tha…

The chance of having an extra fortune in a fortune cookie is about 3 percent. 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.

1. In words, define the random variable X.
2. List the values that X may take on.
3. Give the distribution of X. X ~ _____(…

The chance of an IRS audit for a tax return reporting more than $25,000 in income is about 2 percent per year. Suppose that 100 people with tax returns over $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.

1. In words, define the random variable X.
2. List the values that X may take on.
3. Give the distr…

On average, Pierre, an amateur chef, drops three pieces of eggshell into every two cake batters he makes. Suppose that you buy one of his cakes.

1. In words, define the random variable X.
2. List the values that X may take on.
3. Give the distribution of X. X ~ _____(_____,_____)
4. On average, how many pieces of eggshell do you expect to be in the cake?
5. What is the probability that there will not be any piece…

Use the following information to answer the next two exercises: The average number of times per week that Mrs. Plum’s cats wake her up at night because they want to play is 10. We are interested in the number of times her cats wake her up each week.

Find the probability that her cats will wake her up no more than five times next week.

1. .5000
2. .9329
3. .0378
4. .0671