Mathematics
1.Defective Transistors From past experience, a company has found that in cartons of transistors, 92% contain no defective transistors, 3% contain one defective transistor, 3% contain two defective transistors, and 2% contain three defective transistors. Find the mean, variance, and standard deviation for the defective transistors.
About how many extra transistors per day would the company need to replace the defective ones if it used 10 cartons per day?
2.Suit Sales The number of suits sold per day at a retail store is shown in the table, with the corresponding probabilities. Find the mean, variance, and standard deviation of the distribution.
If the manager of the retail store wants to be sure that he has enough suits for the next 5 days, how many should the manager purchase?
3.Number of Credit Cards A bank vice president feels that each savings account customer has, on average, three credit cards. The following distribution represents the number of credit cards people own. Find the mean, variance, and standard deviation. Is the vice president correct?
4.Trivia Quiz The probabilities that a player will get 5 to 10 questions right on a trivia quiz are shown below. Find the mean, variance, and standard deviation for the distribution.
5.Cellular Phone Sales The probability that a cellular phone company kiosk sells X number of new phone contracts per day is shown below. Find the mean, variance, and standard deviation for this probability distribution.
What is the probability that they will sell 6 or more contracts three days in a row?
6.Animal Shelter Adoptions The local animal shelter adopts out cats and dogs each week with the following probabilities.
Find the mean, variance, and standard deviation for the number of animals adopted each week. What is the probability that they find homes for more than 5 animals in a given week?
7.Commercials During Children’s TV Programs A concerned parents group determined the number of commercials shown in each of five children’s programs over a period of time. Find the mean, variance, and standard deviation for the distribution shown.
8.Number of Televisions per Household A study conducted by a TV station showed the number of televisions per household and the corresponding probabilities for each. Find the mean, variance, and standard deviation.
If you were taking a survey on the programs that were watched on television, how many program diaries would you send to each household in the survey?
9.Students Using the Math Lab The number of students using the Math Lab per day is found in the distribution below. Find the mean, variance, and standard deviation for this probability distribution.
What is the probability that fewer than 8 or more than 12 use the lab in a given day?
10.Pizza Deliveries A pizza shop owner determines the number of pizzas that are delivered each day. Find the mean, variance, and standard deviation for the distribution shown. If the manager stated that 45 pizzas were delivered on one day, do you think that this is a believable claim?
11.Insurance An insurance company insures a person’s antique coin collection worth $20,000 for an annual premium of $300. If the company figures that the probability of the collection being stolen is 0.002, what will be the company’s expected profit?
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12.Job Bids A landscape contractor bids on jobs where he can make $3000 profit. The probabilities of getting 1, 2, 3, or 4 jobs per month are shown.
Find the contractor’s expected profit per month.
13.Rolling Dice If a person rolls doubles when she tosses two dice, she wins $5. For the game to be fair, how much should she pay to play the game?
14.Dice Game A person pays $2 to play a certain game by rolling a single die once. If a 1 or a 2 comes up, the person wins nothing. If, however, the player rolls a 3, 4, 5, or 6, he or she wins the difference between the number rolled and $2. Find the expectation for this game. Is the game fair?
15.Lottery Prizes A lottery offers one $1000 prize, one $500 prize, and five $100 prizes. One thousand tickets are sold at $3 each. Find the expectation if a person buys one ticket.
16.In Exercise 15, find the expectation if a person buys two tickets. Assume that the player’s ticket is replaced after each draw and that the same ticket can win more than one prize.
17.Winning the Lottery For a daily lottery, a person selects a three-digit number. If the person plays for $1, she can win $500. Find the expectation. In the same daily lottery, if a person boxes a number, she will win $80. Find the expectation if the number 123 is played for $1 and boxed. (When a number is “boxed,” it can win when the digits occur in any order.)
18.Life Insurance A 35-year-old woman purchases a $100,000 term life insurance policy for an annual payment of $360. Based on a period life table for the U.S. government, the probability that she will survive the year is 0.999057. Find the expected value of the policy for the insurance company.
19.Raffle Ticket Sales A civic group sells 1000 raffle tickets to raise $2500 for its namesake charity. First prize is $1000, second prize is $300, and third prize is $200. How much should the group charge for each ticket?
Extending the Concepts
20.Rolling Dice Construct a probability distribution for the sum shown on the faces when two dice are rolled. Find the mean, variance, and standard deviation of the distribution.
21.Rolling a Die When one die is rolled, the expected value of the number of spots is 3.5. In Exercise 20, the mean number of spots was found for rolling two dice. What is the mean number of spots if three dice are rolled?
22.The formula for finding the variance for a probability distribution is
σ = Σ[(X – µ)2 · P(X)]
Verify algebraically that this formula gives the same result as the shortcut formula shown in this section.
23.Rolling a Die Roll a die 100 times. Compute the mean and standard deviation. How does the result compare with the theoretical results of Example 5–5?
24.Rolling Two Dice Roll two dice 100 times and find the mean, variance, and standard deviation of the sum of the spots. Compare the result with the theoretical results obtained in Exercise 20.
25.Extracurricular Activities Conduct a survey of the number of extracurricular activities your classmates are enrolled in. Construct a probability distribution and find the mean, variance, and standard deviation.
26.Promotional Campaign In a recent promotional campaign, a company offered these prizes and the corresponding probabilities. Find the expected value of winning. The tickets are free.
Number of prizes | Amount | Probability |
1 | $100,000 | |
2 | 10,000 | |
5 | 1,000 | |
10 | 100 |
If the winner has to mail in the winning ticket to claim the prize, what will be the expectation if the cost of the stamp is considered? Use the current cost of a stamp for a first-class letter.
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Speaking of Statistics
This study shows that a part of the brain reacts to the impact of losing, and it might explain why people tend to increase their bets after losing when gambling. Explain how this type of split decision making may influence fighter pilots, firefighters, or police officers, as the article states.
Source: Psychology Today, August 2002, p. 22. Used with permission.
Technology Step by Step
TI-83 Plus or TI-84 Plus
Step by Step
To calculate the mean and variance for a discrete random variable by using the formulas:
1.Enter the x values into L1 and the probabilities into L2.
2.Move the cursor to the top of the L3 column so that L3 is highlighted.
3.Type L1 multiplied by L2, then press ENTER.
4.Move the cursor to the top of the L4 column so that L4 is highlighted.
5.Type L1 followed by the x2 key multiplied by L2, then press ENTER.
6.Type 2nd QUIT to return to the home screen.
7.Type 2nd LIST, move the cursor to MATH, type 5 for sum, then type L3, then press ENTER.
8.Type 2nd ENTER, move the cursor to L3, type L4, then press ENTER.
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Using the data from Example 5–10 gives the following:
To calculate the mean and standard deviation for a discrete random variable without using the formulas, modify the procedure to calculate the mean and standard deviation from grouped data (Chapter 3) by entering the x values into L1 and the probabilities into L2.
Objective 3
Find the exact probability for X successes in n trials of a binomial experiment.
5–3The Binomial Distribution
Many types of probability problems have only two outcomes or can be reduced to two outcomes. For example, when a coin is tossed, it can land heads or tails. When a baby is born, it will be either male or female. In a basketball game, a team either wins or loses. A true/false item can be answered in only two ways, true or false. Other situations can be reduced to two outcomes. For example, a medical treatment can be classified as effective or ineffective, depending on the results. A person can be classified as having normal or abnormal blood pressure, depending on the measure of the blood pressure gauge. A multiple-choice question, even though there are four or five answer choices, can be classified as correct or incorrect. Situations like these are called binomial experiments.
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Historical Note
In 1653, Blaise Pascal created a triangle of numbers called Pascal’s triangle that can be used in the binomial distribution.
A binomial experiment is a probability experiment that satisfies the following four requirements:
1.There must be a fixed number of trials.
2.Each trial can have only two outcomes or outcomes that can be reduced to two outcomes. These outcomes can be considered as either success or failure.
3.The outcomes of each trial must be independent of one another.
4.The probability of a success must remain the same for each trial.
A binomial experiment and its results give rise to a special probability distribution called the binomial distribution.
The outcomes of a binomial experiment and the corresponding probabilities of these outcomes are called a binomial distribution.
In binomial experiments, the outcomes are usually classified as successes or failures. For example, the correct answer to a multiple-choice item can be classified as a success, but any of the other choices would be incorrect and hence classified as a failure. The notation that is commonly used for binomial experiments and the binomial distribution is defined now.
Notation for the Binomial Distribution
P(S) | The symbol for the probability of success |
P(F) | The symbol for the probability of failure |
p | The numerical probability of a success |
q | The numerical probability of a failure |
P(S) = p and P(F) = 1 – p = q | |
n | The number of trials |
X | The number of successes in n trials |
Note that 0 ≤ X ≤ n and X = 0, 1, 2, 3, … , n.
The probability of a success in a binomial experiment can be computed with this formula.
Binomial Probability Formula
In a binomial experiment, the probability of exactly X successes in n trials is
An explanation of why the formula works is given following Example 5–15.
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Example 5–15
Tossing Coins
A coin is tossed 3 times. Find the probability of getting exactly two heads.
Solution
This problem can be solved by looking at the sample space. There are three ways to get two heads.
HHH, HHT, HTH, THH, TTH, THT, HTT, TTT
The answer is , or 0.375.
Looking at the problem in Example 5–15 from the standpoint of a binomial experiment, one can show that it meets the four requirements.
1.There are a fixed number of trials (three).
2.There are only two outcomes for each trial, heads or tails.
3.The outcomes are independent of one another (the outcome of one toss in no way affects the outcome of another toss).
4.The probability of a success (heads) is in each case.
In this case, n = 3, X = 2, p = , and q = . Hence, substituting in the formula gives
which is the same answer obtained by using the sample space.
The same example can be used to explain the formula. First, note that there are three ways to get exactly two heads and one tail from a possible eight ways. They are HHT, HTH, and THH. In this case, then, the number of ways of obtaining two heads from three coin tosses is 3C2, or 3, as shown in Chapter 4. In general, the number of ways to get X successes from n trials without regard to order is
This is the first part of the binomial formula. (Some calculators can be used for this.)
Next, each success has a probability of and can occur twice. Likewise, each failure has a probability of and can occur once, giving the part of the formula. To generalize, then, each success has a probability of p and can occur X times, and each failure has a probability of q and can occur n – X times. Putting it all together yields the binomial probability formula.
Example 5–16
Survey on Doctor Visits
A survey found that one out of five Americans say he or she has visited a doctor in any given month. If 10 people are selected at random, find the probability that exactly 3 will have visited a doctor last month.
Source: Reader’s Digest.
Solution
In this case, n = 10, X = 3, p = , and q = . Hence,
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Example 5–17
Survey on Employment
A survey from Teenage Research Unlimited (Northbrook, Illinois) found that 30% of teenage consumers receive their spending money from part-time jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them will have part-time jobs.
Solution
To find the probability that at least 3 have part-time jobs, it is necessary to find the individual probabilities for 3, or 4, or 5, and then add them to get the total probability.
Hence,
P(at least three teenagers have part-time jobs)
= 0.132 + 0.028 + 0.002 = 0.162
Computing probabilities by using the binomial probability formula can be quite tedious at times, so tables have been developed for selected values of n and p. Table B in Appendix C gives the probabilities for individual events. Example 5–18 shows how to use Table B to compute probabilities for binomial experiments.
Example 5–18
Tossing Coins
Solve the problem in Example 5–15 by using Table B.
Solution
Since n = 3, X = 2, and p = 0.5, the value 0.375 is found as shown in Figure 5–3.
Figure 5–3
Using Table B for Example 5–18
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Example 5–19
Survey on Fear of Being Home Alone at Night