5.3A Conditional Probability, General

Multiplication Rule and Tree Diagrams

AP Statistics

What is Conditional Probability?The probability that one event happens given that another event is already known to have happened is called a conditional probability.

Suppose we know that event A has happened. Then the probability that event B happens given that event A has happened is denoted by P(B | A).

Read | as “given that” or “under the

condition that”

Calculating Conditional Probabilities

To find the conditional probability P(A | B), use the formula

The conditional probability P(B | A) is given by

Calculating Conditional Probabilities

Calculating Conditional Probabilities

Find P(L)

Find P(E | L)

Find P(L | E)

Total 3392 2952 3656 10000

Total

6300

1600

2100

P(L) = 3656 / 10000 = 0.3656

P(E | L) = 800 / 3656 = 0.2188

P(L| E) = 800 / 1600 = 0.5000

Define events E: the grade comes from an EPS course, and L: the grade is lower than a B.

The General Multiplication RuleThe probability that events A and B both occur can be found using the general multiplication rule

P(A ∩ B) = P(A) • P(B | A)

where P(B | A) is the conditional probability that event B occurs given that event A has already occurred.

General Multiplication Rule

In words, this rule says that for both of two events to occur, first one must occur, and then given that the first event has occurred, the second must occur.

Tree Diagrams

The general multiplication rule is especially useful when a chance process involves a sequence of outcomes. In such cases, we can use a tree diagram to display the sample space.

Consider flipping a coin twice.

What is the probability of getting two heads?

Sample Space:HH HT TH TT

So, P(two heads) = P(HH) = 1/4

Example: Tree Diagrams

51.15% of teens are online and have posted a profile.

The Pew Internet and American Life Project finds that 93% of teenagers (ages 12 to 17) use the Internet, and that 55% of online teens have posted a profile on a social-networking site. What percent of teens are online and have posted a profile?

Tree Diagrams in Reverse: Mammograms Many women choose to have annual mammograms to screen for

breast cancer after age 40. A mammogram isn’t foolproof. Sometimes, the test suggests that a woman has breast cancer when she doesn’t (a false positive); other times, it suggests she doesn’t have breast cancer when she does (a false negative). One percent of the women aged 40 or over in this region have

breast cancer. For women who have breast cancer, the probability of a false

negative is 0.03 For women who don’t have breast cancer, the probability of a

false positive is 0.06.

A randomly selected woman aged 40 or over from this region tests positive for breast cancer in a mammogram. Find the probability that she actually has breast cancer. Show your calculations.

Mammograms: Solution (1/4)

Mammograms: Solution (2/4)

We want to find P(breast cancer│positive mammogram)

P(breast cancer and positive mammogram)P(breast cancer | positive mammogram) =

P(positive mammogram)

To find P(breast cancer and positive mammogram), we use the general multiplication rule along with the tree diagram.

P(breast cancer and positive mammogram) = P(breast cancer) P(positive mammogram | breast cancer)

= (0.01)(0.97) = 0.0097

So we need to calculate a few things first…

Mammograms: Solution (3/4)

To find P(positive mammogram) we have to use both the multiplication rule AND the addition rule.

P(positive mammogram) = (0.01)(0.97) + (0.99)(0.06) = 0.0691(women who have breast cancer and get a positive result) + (women who don't have breast cancer and get a positive result)

Mammograms: Solution (4/4)

P(breast cancer and positive mammogram)P(breast cancer | positive mammogram) =

P(positive mammogram)

NOW we have enough information to do our final calculation.

0.00970.14

0.0691

Given that a randomly selected woman from the region has a positive mammogram, there is about a 14% chance she actually has breast cancer.

Homework

5.3A pg. 317, #57-60 pg. 333-335, #63, 65, 67, 71, 73, 77,

79