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? 1. Find P(own a pet|female). 2. Find P(male|don’t own a pet). YesNo Female86 Male57

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.