1. Section 3.2
Conditional Probability and the
Multiplication Rule

2. Conditional Probability
The probability an event B will occur, given (on the condition) that another event A has occurred.
We write this as P(B|A) and say “probability of B, given A.”
Two cars are selected from a production line of 12 cars where 5 are defective. What is the probability the 2nd car is defective, given the first car was defective?
2 illustrations for conditional probability, one with dependent events the other with independent events.
Given a defective car has been selected, the conditional sample space has 4 defective out of 11. P(B|A) = 4/11

3. Independent Events Two dice are rolled. Find the probability
the second die is a 4 given the first was a 4.
Original sample space: {1, 2, 3, 4, 5, 6}
Given the first die was a 4, the conditional
sample space is: {1, 2, 3, 4, 5, 6}
2 illustrations for conditional probability, one with dependent events the other with independent events. Note that knowing what happens on the first die, does not affect the probability of rolling a 4 on the second.
The conditional probability, P(B|A) = 1/6

4. Independent Events
Two events A and B are independent if the probability of the occurrence of event B is not affected by the occurrence
(or non-occurrence) of event A.
A = Being female
B = Having type O blood
A = 1st child is a boy
B = 2nd child is a boy
Two events that are not independent are dependent.
Intuitive examples for pairs of independent events and pairs of dependent events
A = taking an aspirin each day
B = having a heart attack
A = being a female
B = being under 64” tall

5. Independent Events
If events A and B are independent, then P(B|A) = P(B)
Conditional Probability
Probability
12 cars are on a production line where 5 are defective and 2 cars are selected at random.
A = first car is defective
B = second car is defective.
The probability of getting a defective car for the second car depends on whether the first was defective. The events are dependent.
Distinguish between 2 events that are independent and 2 events that are dependent.
Two dice are rolled. A = first is a 4 and B = second is a 4
P(B) = 1/6 and P(B|A) = 1/6. The events are independent.

6. Contingency Table
The results of responses when a sample of adults in 3 cities was asked if they liked a new juice is:
Omaha
Seattle
Miami
Total
Yes
100
150
150
400 No
125
130 95
350
Undecided 75
170 5
250
Total
300
450
250
1000
One of the responses is selected at random. Find:
A 3 by 3 contingency table
1. P(Yes)
2. P(Seattle)
3. P(Miami)
4. P(No, given Miami)

7. Solutions Omaha Seattle Miami Total Yes 100 150 150 400 No 125 130 95
350
Undecided 75
170 5
250
Total
300
450
250
1000
1. P(Yes)
2. P(Seattle)
3. P(Miami)
4. P(No, given Miami)
= 400 / 1000 = 0.4
Notice how the sample space changes when an event is “given”.
= 450 / 1000 = 0.45
= 250 / 1000 = 0.25
= 95 / 250 = 0.38
Answers: 1) ) ) ) 0.38

8. Multiplication Rule P(A and B) = P(A) x P(B|A) P(A) = 5/12
To find the probability that two events, A and B will occur in sequence, multiply the probability A occurs by the conditional probability B occurs, given A has occurred.
P(A and B) = P(A) x P(B|A)
Two cars are selected from a production line of 12 where 5 are defective. Find the probability both cars are defective.
A = first car is defective B = second car is defective.
P(A) = 5/12
P(B|A) = 4/11
P(A and B) = 5/12 x 4/11 = 5/33 =

9. Multiplication Rule P(A and B) = 1/6 x 1/6 = 1/36 = 0.028
Two dice are rolled. Find the probability both are 4’s.
A = first die is a 4 and B = second die is a 4.
P(A) = 1/6
P(B|A) = 1/6
P(A and B) = 1/6 x 1/6 = 1/36 = 0.028
When two events A and B are independent, then
P (A and B) = P(A) x P(B)
Note for independent events P(B) and P(B|A) are the same.

10. Homework all pgs
Day 2: all pgs