Basic Concepts of Probability

BASIC

1. For two events $A$ and $B, P(A)=0.73, P(B)=0.48$ , and $P(A \cap B)=0.29$ .

a. Find $P(A \mid B)$ .

b. Find $P(B \mid A)$ .

c. Determine whether or not $A$ and $B$ are independent.

3. For independent events $A$ and $B, P(A)=0.81$ and $P(B)=0.27$ .

a. Find $P(A \cap B)$ .

b. Find $P(A \mid B)$ .

c. Find $P(B \mid A)$ .

5. For mutually exclusive events $A$ and $B, P(A)=0.17$ and $P(B)=0.32$ .

a. Find $P(A \mid B)$ .

b. Find $P(B \mid A)$

7. Compute the following probabilities in connection with the roll of a single fair die.

a. The probability that the roll is even.

b. The probability that the roll is even, given that it is not a two.

c. The probability that the roll is even, given that it is not a one.

9. A special deck of 16 cards has 4 that are blue, 4 yellow, 4 green, and 4 red. The four cards of each color are numbered from one to four. A single card is drawn at random. Find the following probabilities.

a. The probability that the card drawn is red.

b. The probability that the card is red, given that it is not green.

c. The probability that the card is red, given that it is neither red nor yellow.

d. The probability that the card is red, given that it is not a four.

11. A random experiment gave rise to the two-way contingency table shown. Use it to compute the probabilities indicated.

	R	S
A	0.12	0.18
B	0.28	0.42

a. $P(A), P(R), P(A \cap R)$

b. Based on the answer to (a), determine whether or not the events $A$ and $R$ are independent.

c. Based on the answer to (b), determine whether or not $P(A \mid R)$ can be predicted without any computation. If so, make the prediction. In any case, compute $P(A \mid R)$ using the Rule for Conditional Probability.

13. Suppose for events $A$ and $B$ in a random experiment $P(A)=0.70$ and $P(B)=0.30$ . Compute the indicated probability, or explain why there is not enough information to do so.

a. $P(A \cap B)$

b. $P(A \cap B)$ , with the extra information that $A$ and $B$ are independent.

c. $P(A \cap B)$ , with the extra information that $A$ and $B$ are mutually exclusive.

15. Suppose for events $A, B$ , and $C$ connected to some random experiment, $A, B$ , and $C$ are independent and $P(A)=0.88, P(B)=0.65$ , and $P(C)=0.44$ . Compute the indicated probability, or explain why there is not enough information to do so.

a. $P(A \cap B \cap C)$

b. $P\left(A^{c} \cap B^{c} \cap C^{c}\right)$

APPLICATIONS

17. The sample space that describes all three-child families according to the genders of the children with respect to birth order is

$\begin{align*} S=\{b b b, b b g, b g b, b g g, g b b, g b g, g g b, g g g\} \end{align*}$

In the experiment of selecting a three-child family at random, compute each of the following probabilities, assuming all outcomes are equally likely.

a. The probability that the family has at least two boys.

b. The probability that the family has at least two boys, given that not all of the children are girls.

c. The probability that at least one child is a boy.

d. The probability that at least one child is a boy, given that the first born is a girl.

19. The following two-way contingency table gives the breakdown of the population in a particular locale according to party affiliation (A, B, C, or None) and opinion on a bond issue:

Affiliation	Opinion
Affiliation	Favors	Opposes	Undecided
A	0.12	0.09	0.07
B	0.16	0.12	0.14
C	0.04	0.03	0.06
None	0.08	0.06	0.03

A person is selected at random. Find each of the following probabilities.

a. The person is in favor of the bond issue.

b. The person is in favor of the bond issue, given that he is affiliated with party A.

c. The person is in favor of the bond issue, given that he is affiliated with party B.

21. The following two-way contingency table gives the breakdown of the population of adults in a particular locale according to employment type and level of life insurance:

Employment Type	Level of Insurance
Employment Type	Low	Medium	High
Unskilled	0.07	0.19	0.00
Semi-skilled	0.04	0.28	0.08
Skilled	0.03	0.18	0.05
Professional	0.01	0.05	0.02

An adult is selected at random. Find each of the following probabilities.

a. The person has a high level of life insurance.

b. The person has a high level of life insurance, given that he does not have a professional position.

c. The person has a high level of life insurance, given that he has a professional position.

d. Determine whether or not the events "has a high level of life insurance" and "has a professional position" are independent.

23. The sensitivity of a drug test is the probability that the test will be positive when administered to a person who has actually taken the drug. Suppose that there are two independent tests to detect the presence of a certain type of banned drugs in athletes. One has sensitivity $0.75$ ; the other has sensitivity $0.85 .$ If both are applied to an athlete who has taken this type of drug, what is the chance that his usage will go undetected?

25. An accountant has observed that $5 \%$ of all copies of a particular two-part form have an error in Part 1 , and $2 \%$ have an error in Part II. If the errors occur independently, find the probability that a randomly selected form will be error-free.

ADDITIONAL EXERCISES

27. Events $A$ and $B$ are mutually exclusive. Find $P(A \mid B)$ .

29. A basketball player makes $60 \%$ of the free throws that he attempts, except that if he has just tried and missed a free throw then his chances of making a second one go down to only $30 \%$ . Suppose he has just been awarded two free throws.

a. Find the probability that he makes both.

b. Find the probability that he makes at least one. (A tree diagram could help.)