13.3 Conditional Probability and Intersections of Events

• Understand how to compute conditional probability.

• Calculate the probability of the intersection of two events.

• Use probability trees to compute conditional probabilities.

• Understand the difference between dependent and independent events.

Conditional probability takes into account that one event occurring may change the probability of a 2nd event.

• When we compute the probability of event F assuming that the event E has already occurred, this is called the conditional probability of F, given E.– We denote this as P(F | E). “The

probability of F given E.”– If E and F are events in a sample space

with equally likely outcomes, then P(F | E)=n(EF)

n(E)

• Assume we roll two dice and the total showing is greater than nine. What is the probability that the total is odd?

– This sample space has 36 equally likely outcomes. Let G stand for the roll being greater than nine and O stand for the total being odd.

– The number of outcomes in G are 6: (4,6), (5,5), (5,6), (6,4), (6,5), (6,6). The number of outcomes of O G are 2: (5,6) and (6,5).

– P(O | G) = n(O G) = 2 = 1 n(G) 6 3

• General Rule for Computing P(F|E): If E and F are events in a sample space, then P(F | E) = P(E F) P(E)– This rule is used when the sample

space is not equally likely or in a situation where it is not possible to count the outcomes. This rule is based on probability rather than counting.

We use conditional probability to find the probability of the intersection of two events.

• Rule for Computing the Probability of the Intersection of Events – If E and F are two events, then P(E F) = P(E) • P(F | E)

• We draw two cards without replacement from a standard 52-card deck. What is the probability that both cards are kings?– P(A B) = P(A) • P(B | A)

4 • 3 = 0.0045 52 51

Trees help you visualize probability computations.

• We can represent an experiment that happens in stages with a tree whose branches represent the outcomes of the experiment. We calculate the probability of an outcome by multiplying the probabilities found along the branch representing that outcome.

Independent events have no effect on each other’s probabilities.

• Events E and F are independent events if P(F | E) = P(F).– If E and F are independent, then

knowing that E has occurred does not influence the way we compute the probability of F

• Events E and F are dependent events if P(F | E) ≠ P(F)

Classwork/Homework

• Classwork – Page 757 (7 – 10, 23 – 35 odd, 41, 43, 67, 69)

• Homework – Page 757 (11 – 14, 24 – 36 even, 42, 44, 68, 70)

Dice Table

1 2 3 4 5 6

1(1,1)

(1,2)

(1,3)

(1,4)

(1,5)

(1,6)

2(2,1)

(2,2)

(2,3)

(2,4)

(2,5)

(2,6)

3(3,1)

(3,2)

(3,3)

(3,4)

(3,5)

(3,6)

4(4,1)

(4,2)

(4,3)

(4,4)

(4,5)

(4,6)

5(5,1)

(5,2)

(5,3)

(5,4)

(5,5)

(5,6)

6(6,1)

(6,2)

(6,3)

(6,4)

(6,5)

(6,6)