AP Statistics Section 6.2C

Independent Events & The Multiplication Rule

Consider a standard deck of 52 playing cards. If one card is chosen at random, what is the probability

the card is red?

2

1

52

26

If the first card is red and a second card is chosen at random from the same deck, the probability that this

card is also red will depend upon what was done with the first card.

If the first card was returned to the deck, the probability the second

card is red will be…

2

1

52

26

If the first card was not returned to the deck, the probability the second card is red will be…

51

25

In the first case, knowing that the first card pulled from the deck was red did not affect the probability

that the second card was red.

This is not true, however, in the second case.

In the second case, knowing that the first card pulled from the deck

was red affected the probability that the second card was red.

Two events are independent if knowing that the one occurs does not affect the probability that the

other one occurs.

Example: Determine if the two events A and B are independent.

independent

Example: Determine if the two events A and B are independent.

dependent

Example: Determine if the two events A and B are independent.

dependent

Example: Are an event A and Ac independent? Why are why not?

Note:

versa.- viceandoccur cannot A

means occured hasA that knowing No,c

Disjoint events are not independent.

Disjoint and independent do not mean the same thing.

Consider rolling a 6-sided die and then flipping a coin. What is the probability of rolling a 3 and

getting a heads on the coin?

12

1

12

1

2

1

6

1 :Note

Multiplication Rule for Independent Events:

If A and B are independent, then

___________________P(B) P(A) B) andP(A

Example: Find the following probabilities.

a. selecting 4 cards with replacement from a standard deck and having all 4 be clubs

256

1

4

1

4

1

4

1

4

1

Example: Find the following probabilities.

b. rolling five 6-sided dice and getting five twos

7776

1

6

1

6

1

6

1

6

1

6

1

Example: Find the following probabilities.

c. selecting 4 cards with replacement from a standard deck and getting at least one club

clubs) P(no1

256

175

256

811

4

3

4

3

3

3

4

31

Example: Find the following probabilities.

d. rolling five 6-sided dice and getting at least one two

twos)P(no1

6

5

6

5

6

5

6

5

6

51

7776

4651

7776

31251