Chapter 3 Exercises

1. If a particular event is defined as a passing score on an examination, what is the complement of the event?

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2. City residents were surveyed recently to determine readership of newspapers. Fifty percent of the residents read the morning paper, 60 percent read the evening paper and 20 percent read both newspapers. Find the probability that a resident selected at random reads either the morning or evening paper, or both.

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3. Consider a job interview situation to be a random experiment. Define two events:

Event A : the candidate had good eye contact
Event B : the candidate got the job.

Assume that
P(A) = 0.40
P(B) = 0.20
P(A and B) = 0.12

Draw a Venn diagram to summarise the situation.

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4. Suppose the table below shows the distribution of colours of M&Ms.

Colour	Brown	Red	Yellow	Green	Orange	Tan
Probability	0.3	0.2	0.2	0.1	0.1	0.1

Find the probabilities of each of the following:

(i)You select brown or red.
(ii)You select green, red or tan.
(iii)The M&M you select is not yellow.
(iv)The M&M you select is neither orange nor tan.

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5. A restaurant has collected data on its customers orders and so has estimated empirical probabilities of what happens after the main course. It was found that 20% had dessert only, 40% had coffee only, and 30% had both dessert AND coffee.

(a) Draw a Venn diagram for this situation.
(b) Find the probability of the event "had coffee". (Hint: Be careful; this event includes those who did as well as those who did not have dessert).
(c) Find the probability of the event "did NOT have dessert".
(d) Find the probability of the event "neither coffee nor dessert".
(e) Find the probability of the event "had coffee OR dessert".
(f) Are the events "had coffee" and "had dessert" mutually exclusive? How do you know?
(g) Find the conditional probability of ordering coffee GIVEN that the customer ordered dessert.
(h) Are "had dessert" and "had coffee" independent events? How do you know?
(i) Find the conditional probability of ordering dessert GIVEN that the customer ordered coffee.
(j) Find the conditional probability of ordering dessert GIVEN that the customer did not order coffee.
(k) To see if coffee and dessert seem to go well together, compare your answers to parts (i) and (j) above. In particular, who is more likely to order dessert: a customer who orders coffee or one who does not?

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6. The following table shows the joint probability (relative frequency) distribution for the type and size of hospital in a particular region.

Type of Hospital
		General	Teaching
Size	Small	0.500	0.084	0.584
	Large	0.313	0.103	0.416
		0.813	0.187	1.000

If a hospital in the region is chosen at random what is the probability that it is:
(a) a teaching hospital
(b) a large teaching hospital
(c) a large hospital, given that it is a teaching hospital
(d) a teaching hospital, given that it is a large hospital.

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7. For the data in Question 3, Calculate

(i) P(A|B)
(ii) P(B|A)

In each case, explain in words what probability you are calculating. Does good eye contact enhance the chance of success?

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8. An appliance dealer offers a three-year service contract with each appliance sold. Approximately 40 percent of the customers purchase service contracts. Twenty percent of the sales are for air conditioners. In the past, about 25 percent of those purchasing service contracts were air conditioner purchasers.

(a) What is the probability that a customer purchases an air conditioner and service contract ?
(b) If the next customer buys an air conditioner, what are the chances he or she will want the service contract ?

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9. Only 15% of novels written are ever published. 50% of those published have a happy ending, while 79% of those never published have a happy ending. If a newly written novel has a happy ending, what are its chances of publication?

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10. Determine whether each of the following variates is discrete or continuous:

1. The number of defective ball point pens in each carton of twelve
2. The number of interruptions (breakdowns) per day at a computer facility
3. The distance required for stopping an automobile travelling at 40 kph
4. The number of commercial loans processed per day at a bank
5. The volume of orange juice in each one-litre container.

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11. Suppose the distribution of numbers of people per household is as follows:

Number	1	2	3	4	5	>5
Probability	0.24	0.32	0.18	0.16	0.07	x

1. What is the mean of X ?
2. What is the probability of fewer than 3 people per household ?
3. What is the probability of between 2 and 4 people per household (inclusive) ?
4. What is the probability of at least 4 people per household ?

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12. In each case below, is it reasonable to use a binomial distribution for the random variable X ? Give reasons for your answer in each case.

(a) A car manufacturer chooses one car from each hourUs production for a det ailed quality inspection. One variable recorded is the count X of finish defe cts (dimples, ripples, etc) in the car's paint.
(b) The pool of potential jurors for a murder case contain 100 persons chosen at random from the adult residents of a large city. Each person in the pool is asked whether he or she opposes the death penalty; X is the number who say "Yes".
(c) Joe buys a lottery ticket every week; X is the number of times in a year that he wins a prize.

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13. Evaluate the following:

(a) 5!

(b) 8! / 4!

(c)

(d)

(e)

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14. The survival rate during a risky operation for patients with no other hope of survival is 80%. What is the probability that exactly four of the next five patients survive this operation?

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15. According to government data, 25% of employed women have never been married.

(a) If 10 employed women are selected at random, what is the probability that exactly 2 have never been married ?
(b) What is the probability that 2 or fewer have never been married ?
(c) What is the probability that more than 2 have never been married ?

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16. A sales representative for a computer company contacts five clients each month in an attempt to sell each a new computer system. Data collected over many months has resulted in the following relative frequency distribution:

Number sold per month	0	1	2	3	4	5
Relative frequency	0.05	0.12	0.25	0.30	0.20	0.08

Assuming that this distribution is the probability distribution of the random variable X, the number of systems sold in a month,
(a) What is the probability in a given month of selling 3 or more new computer systems ?
(b) What is the probability of selling less than 2 new computer systems ?
(c) Calculate the expected value, variance and standard deviation of X.

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17. According to government data, 22% of American children under the age of 6 live in households with incomes less than the official poverty level. A random sample of 15 children is selected for a study of learning in early childhood.

(a) Calculate the probability that exactly 3 children in the sample come from poverty-level households.
(b) What is the mean number of children in such a sample who come from poverty-level households? What is the standard deviation of this number between repeated samples?

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18. Using the information in the previous question, now consider a random sample of 300 children.

(a) What is the mean number of children in the sample who come from poverty-level households ? What is the standard deviation of this number?
(b) Use the normal approximation to calculate the probability that

1. at least 80 of the children in the sample live in poverty
2. less than 75 of the children in the sample live in poverty (use the continuity correction).

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