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Chapter 5: Problem 85

A household receives an average of \(1.7\) pieces of junk mail per day. Find the probability that this household will receive exactly 3 pieces of junk mail on a certain day. Use the Poisson probability distribution formula.

Short Answer

Expert verified
Therefore, the probability that the household will receive exactly 3 pieces of junk mail on a certain day is approximately 0.149.

Step by step solution

01

Identifying the given values

From the problem, we can see that the average rate (位) at which the event happens is 1.7 per day (junk mails), and the actual number of successes \(x\) we want to find the probability for is 3 (junk mails).
02

Plugging in the values in the formula

Substitute the values into the Poisson probability distribution formula, we get \( P(3; 1.7) = (e^{-1.7} * 1.7鲁) / 3! \).
03

Calculating the e to the power of -位

Calculate the negative exponential term \(e^{-1.7}\) with the use of a calculator or any online tool, you would get approximately 0.1827.
04

Calculating the 位 to the power of x

Calculate \(1.7^3\), which comes out to be about 4.913.
05

Calculating x factorial

Calculate \(3!\), it means \(3 * 2 * 1\), which is 6.
06

Compute the probability

Now complete the calculation by multiplying the results from steps 3 and 4, then dividing by the result from step 5. Hence, \( P(3; 1.7) = (0.1827 * 4.913) / 6 \), which is approximately 0.149.

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Most popular questions from this chapter

An instant lottery ticket costs \(\$ 2\). Out of a total of 10,000 tickets printed for this lottery, 1000 tickets contain a prize of \(\$ 5\) each, 100 tickets have a prize of \(\$ 10\) each, 5 tickets have a prize of \(\$ 1000\) each, and 1 ticket has a prize of \(\$ 5000 .\) Let \(x\) be the random variable that denotes the net amount a player wins by playing this lottery. Write the probability distribution of \(x\). Determine the mean and standard deviation of \(x\). How will you interpret the values of the mean and standard deviation of \(x ?\)

Briefly explain the concept of the mean and standard deviation of a discrete random variable.

Let \(x\) be a discrete random variable that possesses a binomial distribution. Using the binomial formula, find the following probabilities. a. \(P(x=0)\) for \(n=5\) and \(p=.05\) b. \(P(x=4)\) for \(n=7\) and \(p=.90\) c. \(P(x=7)\) for \(n=10\) and \(p=.60\) Verify your answers by using Table I of Appendix \(\mathrm{C}\).

Residents in an inner-city area are concerned about drug dealers entering their neighborhood. Over the past 14 nights, they have taken turns watching the street from a darkened apartment. Drug deals seem to take place randomly at various times and locations on the street and average about three per night. The residents of this street contacted the local police, who informed them that they do not have sufficient resources to set up surveillance. The police suggested videotaping the activity on the street, and if the residents are able to capture five or more drug deals on tape, the police will take action. Unfortunately, none of the residents on this street owns a video camera and, hence, they would have to rent the equipment. Inquiries at the local dealers indicated that the best available rate for renting a video camera is \(\$ 75\) for the first night and \(\$ 40\) for each additional night. To obtain this rate, the residents must sign up in advance for a specified number of nights. The residents hold a neighborhood meeting and invite you to help them decide on the length of the rental period. Because it is difficult for them to pay the rental fees, they want to know the probability of taping at least five drug deals on a given number of nights of videotaping. a. Which of the probability distributions you have studied might be helpful here? b. What assumption(s) would you have to make? c. If the residents tape for two nights, what is the probability they will film at least five drug deals? d. For how many nights must the camera be rented so that there is at least \(.90\) probability that five or more drug deals will be taped?

Which of the following are binomial experiments? Explain why. a. Drawing 3 balls with replacement from a box that contains 10 balls, 6 of which are red and 4 are blue, and observing the colors of the drawn balls b. Drawing 3 balls without replacement from a box that contains 10 balls, 6 of which are red and 4 are blue, and observing the colors of the drawn balls c. Selecting a few households from New York City and observing whether or not they own stocks when it is known that \(28 \%\) of all households in New York City own stocks
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