Section 11.8Conditional Probability

Math in Our World

Learning Objectives

Find conditional probabilities.

5 yellows & 3 Blues

Conditional Probability

( and )( )( )

P A BP B AP A

We know that to find the probability of two dependent events occurring, it’s important to find the probability of the second event occurring given that the first has already occurred. We call this the conditional probability of event B occurring given that event A has occurred, and denote it P(B | A).

EXAMPLE 7 Finding a Conditional Probability

Suppose that your professor goes stark raving mad and chooses your final grade from A, B, C, D, F, or Incomplete totally at random. Find the probability of getting an A given that you get a letter grade higher than D.

EXAMPLE 7 Finding a Conditional Probability

Suppose that your professor goes stark raving mad and chooses your final grade from A, B, C, D, F, or Incomplete totally at random. Find the probability of getting an A given that you get a letter grade higher than D.

We are asked to find P(A | letter grade higher than D).Method 1 Knowing that you got a letter grade higher than D reduces the sample space to {A, B, C}, which has three outcomes. One of them is an A, so

P(A | letter grade higher than D) =

SOLUTION

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EXAMPLE 7 Finding a Conditional Probability

Method 2 With the full sample space of {A, B, C, D, F, I},SOLUTION

1(A)6

P 3(letter grade higher than D)6

P

16( letter grade higher than D)36

P A 1 66 3

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EXAMPLE 8 Finding a Conditional Probability

Hate crimes are defined to be crimes in which the victim is targeted because of one or more personal characteristics, such as race, religion, or sexual orientation. The table below lists the motivation for certain crimes as reported by the FBI for 2007.

(a) Find the probability that a hate crime was racially motivated given that it was a crime against persons.

Motivation Crimes against persons

Crimes against property

Crimes against society

Race 3,031 1,686 7Religion 421 1,054 2Sexual Orientation 1,039 418 3Total 4,491 3,158 12

Motivation Crimes against persons

Crimes against property

Crimes against society

Race 3,031 1,686 7Religion 421 1,054 2Sexual Orientation 1,039 418 3Total 4,491 3,158 12

EXAMPLE 8 Finding a Conditional Probability

(a) Since we are interested only in crimes against persons, we only need to look at that column. There were 4,491 such crimes total, and 3,031 were racially motivated, so the probability is

SOLUTION

EXAMPLE 8 Finding a Conditional Probability

Hate crimes are defined to be crimes in which the victim is targeted because of one or more personal characteristics, such as race, religion, or sexual orientation. The table below lists the motivation for certain crimes as reported by the FBI for 2007.

(b) Find the probability that a hate crime was against property given that it was motivated by the victim’s sexual orientation.

Motivation Crimes against persons

Crimes against property

Crimes against society

Race 3,031 1,686 7Religion 421 1,054 2Sexual Orientation 1,039 418 3Total 4,491 3,158 12

Motivation Crimes against persons

Crimes against property

Crimes against society

Race 3,031 1,686 7Religion 421 1,054 2Sexual Orientation 1,039 418 3Total 4,491 3,158 12

EXAMPLE 8 Finding a Conditional Probability

(b) This time we are given that the crime was motivated by sexual orientation, so we only need to look at that row. There were 1,460 such crimes total, and 418 were against property, so the probability is

SOLUTION