Probability Distributions (79 problems)

Law Enforcement: Property Crime. Does crime pay? The FBI Standard Survey of Crimes showed that for about 80% of all property crimes, the criminals are never found and the case is never solved. Suppose that a neighborhood district in a large city has repeated property crimes, not always by the same criminals. The police are investigating 6 property crimes in this district

(a) What is the probability that none of the crimes will ever be solved?

(b) What is the probability that at least one crime will be solved?

(c) What is the expected number of crimes that will be solved? What is the standard deviation?

Four cards are selected, one at a time, from a standard deck of 52 cards. Let x represent the number of aces drawn in a set of 4 cards.

1. If this experiment is completed without replacement, explain why x is not a binomial random variable.

2. If this experiment is completed with replacement, explain why x is a binomial random variable.

The probability that an individual is left-handed is 0.1. In a class of 30 students, what is the probability of finding at least five left-handers?

Focus Problem: Personality Types. We now have tools to solve Chapter Focus problem. In the book A Guide to the Development and Use of the Myers-Briggs Type Indicators, by Myers and McCaully, it was reported that approximately 45% of all university professors are extroverted. Suppose that you have classes with 6 professors

(a) What is the probability that all six are extroverts?

(b) What is the probability that none of them is an extrovert?

(c) What is the probability that at least 2 of your professors are extroverts.

(d) In a group of 6 professors selected at random, what is the expected number of extroverts?

Twenty-five school children are checked for head lice. The public health department says that the probability of any given child having lice is 0.1. Let x be the number of children who have head lice.

a) What is the probability that exactly 3 children have head lice?

b) What is the probability that 5 or less children have head lice?

c) What is the probability distribution for x?

d) What is the probability that the number of children who are found to have head lice is within one standard deviation of the mean? Two standard deviations of the mean?

e) Based on part d), what can you say about the shape of the distribution?

Construct the probability distribution for the sum shown on the faces when two dice are rolled.

Business Ethics: Privacy Are your finances, buying habits, medical records and phone calls really private? A real concern for many adults is that computers Internet are reducing privacy. A survey conducted by Peter D. Hart Research Associates for the Shell Poll was reported in USA Today. According to the sun-et_ of adults are concerned that employers are monitoring phone calls. Use Bino­mial distribution formula to calculate the probability that

a.) out of 5 adults, none is concerned that employers are monitoring phone calls

b.) out of 5 adults, all are concerned that employers are monitoring phone calls

c.) out of 5 adults, exactly 3 are concerned that employers are monitoring phone calls

57% of families say that their children have an influence on their vacation destinations. Consider a random sample of six families who are asked if their children have an influence on their vacation destination.

(a) Construct a binomial distribution

(b) Find the mean, variance and standard deviation of the distribution

Naturalists have discovered that the rate of deformation in fish in Lake Ontario is 4 per 100,000. A catch-and-release expedition is being mounted to observe fish deformations.

a) What is the probability that 1 deformed fish per 100,000 will be found?

b) What is the probability that 5 or more deformed fish per 100,000 will be found?

c) What is the probability that the number of deformed fish found will be either 3, 4, or 5?

The college health center did a campus-wide survey of students and found that 15% of the students smoke cigarettes. A group of nine students randomly come together and sit at the same table on the plaza in front of the library. Find the probability that:

a) No student at the table smokes.

b) At least one student at the table smokes.

c) More than two students smoke.

d) From one to five smoke (including one and five).

The manager of a restaurant knows from experience that 70% of the people who make reservations for the evening show up for dinner. The manager decides one evening to overbook and accept 20 reservations when only 15 tables are available. What is the probability that more than 15 parties will show up?

A study conducted at a certain college shows that 53% of the school's graduates find a job in their chosen field within a year after graduation. Find the probability that among 6 randomly selected graduates, at least one finds a job in his or her chosen field within a year of graduating.

DiGiorno’s frozen pizza has some of the most creative and likeable advertisements on television. USA Today’s Ad Track claims that 20% of viewers like the ads “a lot.” Suppose that a sample of 400 television viewers is shown the advertisements. What is the probability that the sample will have between

a. 18% and 22% who like the ads “a lot”?

b. 16% and 24% who like the ads “a lot”?

c. 14% and 26% who like the ads “a lot”?

d. 12% and 28% who like the ads “a lot”?

A die is rolled 12 times. Find the probabilities of rolling exactly 2 ones. (Hint: Use the Binomial Probability formula)

Quality Control: Syringes The quality-control inspector of a production plant will reject a batch of syringes if two or more defective syringes are found in a random sample of eight syringes taken from the batch. Suppose the batch contains 1% defec­tive syringes.

a.) Make a histogram showing the probabilities of r = 0, 1, 2, 3, 4, 5, 6, 7, or 8 defective syringes in a random sample of eight syringes.

b.) Find. What is the expected number of defective syringes the inspector will find?

c.) What is the probability that the batch will be accepted?

d.) Find.

A test consists of 10 true/false questions. To pass the test a student must answer at least 6 questions correctly. If a student guesses on each question, what is the probability that the student will pass the test?

Binomial Probability

A family has planted 15 oak tree saplings in their backyard. Their garden centre has informed them that that the probability that any given sapling will thrive is 0.8. Let x be the number of saplings that thrive.

a) What is the probability that exactly 8 saplings will thrive?

b) What is the probability that less than 12 saplings will thrive?

c) What is the probability that 10 or more saplings will thrive?

d) What is the mean for this probability distribution? What is the probability that exactly this number of saplings will thrive?

Sociology: Ethics The one-time fling! Have you ever purchased an article of clods (dress, sports jacket, etc.), worn the item once to a party, and then returned the chase? This is called a one-time fling. About 10% of all adults deliberately do a time fling and feel no guilt about it! (Source: Are You Normal? by Bernice Kane St. Martin's Press.) In a group of seven adult friends, what is the probability that

(a) no one has done a one-time fling?

(b) at least one person has done a one-time fling?

(c) no more than two people have done a one-time fling?

A surgeon has a 92% success rate for a certain operation. What is the probability that at least 6 of his next 7 patients will survive?