2.8.1. Conditional probability

Definition:

Conditional probability is the probability that an event will occur given that another event has already occurred. If A and B are two events, then the conditional probability of A is written as and read as “the probability of A given that B has already occurred”.

If A and B are two events, then

and

given that and .

Example:

A box contains black chips and white chips. A person selects two chips without replacement. If the probability of selecting a black chip and a white chip is 15/56, and the probability of selecting a black chip on the first draw is 3/8, find the probability of selecting the white chip on the second draw, given that the first chip selected was a black chip.

Solution:

Let B=selecting a black chip

W=selecting a white chip.

Then .

Hence, the probability of selecting a white chip on the second draw given that the first chip selected was black is 5/7.

Example:

In a certain region of Kazakhstan, the probability that a person lives at least 80 years is 0.75 and the probability that he or she lives at least 90 years is 0.63. What is the probability that randomly selected 80-year old person from this region will survive to become 90?

Solution:

Let A and B be the events that the person selected survives to become 90

and 80 years old, respectively. We are interested in . By definition,

(Note that in this case ).

2.8.2. The multiplication rule of probability

Let A and B be two events. The probability of their intersection is

Also

Example:

Suppose that seven nondefective and three defective goods have been mixed up. To find defective goods, we test them one by one, at random, and without replacement. What is the probability that we are lucky and find both of the defective goods in the first two tests?

Solution:

Let and be the events of finding defective goods in the first and second tests respectively. We are interested in

As we know, there are three defective goods in total 10 goods. Consequently, the probability of selecting a defective good at the first selection is . To calculate the probability , we know that the first good is defective because has already occurred. Because the selections are made without replacement, there are 9 total goods and 2 of them are defective at the time of the second selection. Therefore =2/9. Hence the required probability is

= .

Remark: Multiplication rule can be generalized for calculating the probability of the joint occurrence of several events.

For example, if ,then

2.8.3. Multiplication rule for independent events

Two events A and B are independent if

Equivalent conditions are or .

Example:

An urn contains three red balls, two blue balls, and five white balls. A ball is selected and its colour is noted. Then it is replaced. A second ball is selected and its colour is noted. Find the probability of

a) Selecting two blue balls

b) Selecting a blue and then white ball

c) Selecting a red ball and then a blue ball

Solution:

a) P (blue and blue)=P(blue)· P(blue)=

b) P (blue and white)=P(blue)· P(white)=

c) P (red and blue)=P(red)· P(blue)=

Example:

An urn contains five red and seven blue balls. Suppose that two balls are selected at random with replacement. Let A and B be the events that the first and the second balls are red, respectively. Then we get . Now since and .

Thus A and B are independent.

If we do the same experiment without replacement, then

while

as expected. Thus implying that A and B are dependent.

Remark:

Multiplication rule for independent events can also be extended to three or more independent events by using the formula

.

Example:

The probability that a specific medical test will show positive is 0.32. If four people are tested, find the probability that all four will show positive.

Solution:

Let (i=1, 2, 3, 4) be the symbol for a positive test result.

Exercises

1. Suppose that , , and

a) Find .

b) Are the events A and B independent? Why or why not?

2. Suppose that , , and . Find

a) The conditional probability that B occurs, given that A occurs.

b) The conditional probability that B does not occur given that A occurs.

c) The conditional probability that B occurs given that A does not occur.

3. Concerning the events A and B, the following probabilities are given

; ; .

Determine

a) ; b) ; c)

4. In a study of television viewing habits among married couples, a researcher found that for a popular Saturday night program 25% of the husbands viewed the program regularly and 30% of the wives viewed the program regularly. The study found that for couples where the husband watches the program regularly 80% of the wives also watch regularly.

a) What is the probability that both husband and wife watch the program regularly?

b) What is the probability that at least one-husband or wife-watches the program regularly?

c) What percentage of married couples do not have at least one regular viewer of the program?

5. Of 20 rats in a laboratory, 12 are males and 9 are infected with a virus. Of the 12 male rats, 7 infected with the virus. One rat is randomly selected from the laboratory.

a) If the selected rat is found to be infected, what is the probability that it is a female?

b) If the selected rat is found to be a male, what is the probability that it is infected?

c) Are the events “the selected rat is infected” and “the selected rat is male” independent? Why or why not?

6. Suppose , .

a) Determine if A and B are independent.

b) Determine if A and B are mutually exclusive.

c) Find if A and B are mutually exclusive.

7. An urn has three red and five blue balls. Suppose that 8 balls are selected at random and with replacement. What is the probability that the first three are red and the rest are blue balls?

Answer

1. a) ; ; =0.67; b) No;

2. a) 0.471; b) 0.529; c) 0.719; 3. a) 2/7; b) 32/63; c) 9/16; 4. a) 0.2;

b) 0.35; c) 65%; 5. a) 2/9; b) 7/12; c) No; 6. a) 0.61; b) 0.72; c) 0.641;

7. 0.00503;