1. Multiple Choice Practice
Probability Review

2. D The following is from a particular region’s mortality table.
That is the probability that a 20 year-old will survive to be 60?
(a) (b) (c)
(d) (e)
Age 20 40 60 80
Number Surviving
10,000
9,700
9,240
7,800
4,300
7800/9700 = .8041 D

3. A travel agent books passages on three different tours, with half her customers choosing tour T1, one- third choosing T2, and the rest choosing T3. The agent has noted that three-quarters of those who take tour T1 return to book passage again, two-thirds of those who take T2 return, and one half of those who take T3 return. If a customer does return, what is the probability that the person first went on tour T2?
(a) 1/ (b) 2/ (c) 2/9
(d) 16/ (e) 49/72
P(T3) = 1 – (1/2 + 1/3) = 1/6
P(return) = 3/8 + 2/9 + 1/12 = 49/72
P(T2|return) = (2/9) / (49/72) = 16/49 D

4. If P(A) = 0.25 and P(b) = 0.34, what is P(AUB) if A and B are independent?
(a) (b) (c) (d) .675
(e) There is insufficient info to answer this question
If A and B are independent than P(A and B) = P(A)PB) so
P(A U B) = –(.25)(.34)= .505 B

5. Suppose that the probabilities that an answer can be found Google is
Suppose that the probabilities that an answer can be found Google is .95, on Answers.com is .92, and on both Web sites is Are the possibilities of finding the answer on the two web sites independent?
(a) Yes, because (.95)(.92) = .874.
(b) No, because (.95)(.92) = .874.
(c) Yes, because .95 > .92 > .874.
(d) No, because .5( ) ≠ .874.
(e) There is insufficient information to answer this question A
If P(E and F) = P(E)P(F), then E and F are independent.

6. There are five outcomes to an experiment and a student calculates the respective probabilities of the outcomes to be .34, .50, .42, 0, and The proper conclusion is that
The sum of the individual probabilities is 1.
One of the outcomes will never occur.
One of the outcomes will occur 50 percent of the time.
All of the above are true.
The student made an error.
Probabilities are never less than 0!!! E

7. There are two games involving flipping a fair coin
There are two games involving flipping a fair coin. In the first game, you win a prize if you can throw between 45 percent and 55 percent heads; in the second game, you win if can throw more than 60 percent heads. For each game, would you rather flip the coin 30 times or 300 times?
(a) 30 times for each game
(b) 300 times for each game
(c) 30 times for the first game, and 300 for the second
(d) 300 times for the first game, and 30 for the second
(e) The outcomes of the games do no depend on the number of flips. D
The probability of throwing a head is .5. By the Law of Large Numbers, the more times your flip a coin, the closer the relative frequency tends to be to this probability. With fewer tosses, there is no more chances of wide swings in the relative frequency.

8. Given that 49. 0 percent of the US population are male, and 12
Given that 49.0 percent of the US population are male, and 12.1 percent of the population are over 65 years of age, can we conclude that (.490)(.121)=5.93 percent of the population are men older than 65?
(a) Yes, by the multiplication rule
(b) Yes, by conditional probabilities
(c) Yes, by the Law of Large Numbers
(d) No, because the events are not independent
(e) no, because the events are not mutually exclusive D
P(E and F) = P(E)P(F) only if E and F are independent. In this case, men do not live as long as women, and so the events are not independent.

9. According to one poll, only 8 percent of the public say they “trust Congress.” In a simple random sample of ten people, what is the probability that at least one person “trusts Congress”
(a) .188
(b) .378
(c) .434
(d) .566
(e) .622 D
P(at least one) = 1 –P(none) = 1 – (.92)10 = = .566

10. Suppose you toss a fair coin times and it comes up heads every time
Suppose you toss a fair coin times and it comes up heads every time. Which of the following is a true statement?
(a) By the Law of Large Numbers, the next toss is more likely to be tails than another heads.
(b) By the properties of conditional probability, the next toss is more likely to be heads given that ten tosses in a row have been heads.
(c) Coins actually do have memories, and thus what comes up on the next toss is influenced by the past tosses.
(d) The Law of Large Numbers tells many tosses will necessary before the percentages of heads are again in balance.
(e) None of the above are true statements.
Coins have no memory, and so the probability that the next toss will be heads is .5, and the probability that it will be tails is .5. The Law of Large Numbers says that as the number of tosses becomes larger, the percentages of heads tends to become closer to .5. E

11. C Suppose P(x) = .25 and P(Y) = .40. If P(X|Y) = .20, what is P(Y|X)?
(b) .125
(c) .32
(d) .45
(e) .50 C
P(X|Y)=P(X and Y)/P(Y)
.2 = P(X and Y/ .4
.08 = P(X and Y)
P(Y|X) = P(Y and X)/P(X)
= 0.8 /.24
= .32

12. Given the probabilities P(A) =. 3 and P(AUB)=
Given the probabilities P(A) = .3 and P(AUB)= .7, what is the probabilities P(B) if A and B are mutually exclusive? If A and B are independent?
(a) .4, .3
(b) .4, 4/7
(c) 4/7, .4
(d) .7, 4/7
(e) .7, .3 B
If A and B are mutually exlusive, P(A and B) = 0, so…
.7 = .3 + P(B) – 0, and so P(B) = .4
If A and B are independent, than P(A and B) = P(A)P(B). So, .7 = .3 + P(B) - .3P(B), and so P(B) = 4/7

13. A basket ball player makes one out of his first two free throws
A basket ball player makes one out of his first two free throws. From that point on, the probability that he makes the next shot is equal to the proportion of shots made up to that point. If he makes two more shots, what is the probability he ends up making a total of two free throws?
(a) ¼ (b) 1/3 (c) ½
(d) 2/3 (e) 3/4
Makes 3rd shot
Misses 3rd shot
Makes shot
Misses shot
1/2
2/3
1/3
P(2 shots made) = (1/2)(1/3) + (1/2)(1/3) = 1/3 B

14. Which of the following is not a random variable?
The roll of a fair die
The temperature in Vancouver, BC
The number of votes for different candidates in an election
The number of minutes in a hour
The heart rates of patients in Long Beach Memorial Hospital
ANSWER: D (60 minutes is a constant)

15. Which value completes the probability distribution below?
2/15
13/103
1/5
17/20
ANSWER: E X 1 2 3 4 5
P(X)
1/5
1/6
1/8
Since the probabilities must sum to 1, let x be the unknown value.
1= 1/5+x+1/5+1/6+1/6+1/8.
Solve for x, P(1) = 17/20

16. The distribution cannot be determined.
Suppose you have an unfair die, one weight so that the even numbers have twice the probability distribution. Which statements are true?
The distribution cannot be determined.
This is a continuous distribution.
(a) I only (d) II only
(b) I and III only (e) III only
(c) I and II only
ANSWER: D X 1 2 3 4 5 6
P(X)
Let x represent to probability of the odd values. The 2x would be the probability of the even values. Since the probabilities must sum to 1.
1 = x+ 2x + x + 2x + x + 2x = 9x
Therefore,
x = 1/9 which is the probability of the odd values and the probability of the even values is twice that, or 2/9

17. Which of the following best describes random variables? I.
II. A value, discrete or continuous, that is assigned as an outcome of a random experiment
III. A value, never continuous, that is an outcome of a random experiment
(a) I only (d) I and II only
(b) I and III only (e) III only
(c) II only
ANSWER: C
II only. Random variables are both discrete and continuous. They are outcomes of a random experiment.