Conditional Probability

Finding Conditional Probability

 Definition 1: A conditional probability contains a condition that may limit the sample space for an event. You can write a conditional probability using the notation P(B | A), read “the probability of event B, given event A.”

Example 1

 The table shows the results of a class survey.

 A. Find P(did a chore | male).

 B. Find P(female | did a chore).



Example 2

Recycle Americans recycle increasingly more materials through municipal waste collection each year. The table shows recycling data for a recent year. Find the probability that a sample of recycled waste was paper.

 Find the probability that a sample of recycled waste was paper.

 Find the probability that a sample of recycled waste was plastic.

Using Formulas and Tree Diagrams

 Property: Conditional Probability Formula:  For any two events A and B from a sample space with P(A) ≠ 0.

Example 3

 Market Research Researchers asked shampoo users whether they apply shampoo directly to the head, or indirectly using a hand. Find the probability that a respondent applies shampoo directly to the head, given that the respondent is female.

P(directly to head|female) =

Ticket Out the Door

 The table below shows the results of a class survey.

Do You Own a Pet?

Yes No

Female Male 8 5 6 7 1. Find P(own a pet|female).

2. Find P(male|don’t own a pet).

Tree Diagrams

Example 4

 A student in Buffalo, New York, made the observations below.    Of all snowfalls, 5% are heavy (at least 6 in.).

After a heavy snowfall, schools are closed 67% of the time.

After a light (less than 6 in.) snowfall, schools are closed 3% of the time.

 Find the probability that the snowfall is light and the schools are open.

 Make a tree diagram. Use H for heavy snowfall, L for light snowfall, C for schools closed, and O for schools open.

Example 4 Continued

 a. Find P(L and O)  b. Find P(Schools open, given heavy snow)

Example 5

 Make a tree diagram based on the survey results below. Then find P(a female respondent is left-handed) and P(a respondent is both male and right-handed).

   Of all the respondents, 17% are male.

Of the male respondents, 33% are left-handed.

Of female respondents, 90% are right-handed.

 P(female is left-handed) =  P(both male and right-handed) =

Ticket Out the Door

 A student made the following observations of the weather in his hometown.  On 28% of the days, the sky is mostly clear.  During the mostly clear days, it rained 4% of the time.

 During the cloudy days, it rained 31% of the time.

 Use a tree diagram to find the probability that a day will start out clear, and then it will rain.