### Learning Outcomes

- Describe a sample space and simple and compound events in it using standard notation
- Calculate the probability of an event using standard notation
- Calculate the probability of two independent events using standard notation
- Recognize when two events are mutually exclusive
- Calculate a conditional probability using standard notation

In the previous section we computed the probabilities of events that were independent of each other. We saw that getting a certain outcome from rolling a die had no influence on the outcome from flipping a coin, even though we were computing a probability based on doing them at the same time.

In this section, we will consider events that *are *dependent on each other, called **conditional probabilities**.

### Conditional Probability

The probability the event *B* occurs, given that event *A* has happened, is represented as

*P*(*B* | *A*)

This is read as “the probability of *B* given *A*”

For example, if you draw a card from a deck, then the sample space for the next card drawn has changed, because you are now working with a deck of 51 cards. In the following example we will show you how the computations for events like this are different from the computations we did in the last section.

### example

What is the probability that two cards drawn at random from a deck of playing cards will both be aces?

### Conditional Probability Formula

If Events *A* and *B* are not independent, then

*P*(*A* and *B*) = *P*(*A*) · *P*(*B* | *A*)

### example

If you pull 2 cards out of a deck, what is the probability that both are spades?

### Try It

### Example

The table below shows the number of survey subjects who have received and not received a speeding ticket in the last year, and the color of their car. Find the probability that a randomly chosen person:

- has a speeding ticket
*given*they have a red car - has a red car
*given*they have a speeding ticket

Speeding ticket | No speeding ticket | Total | |

Red car | 15 | 135 | 150 |

Not red car | 45 | 470 | 515 |

Total | 60 | 605 | 665 |

These kinds of conditional probabilities are what insurance companies use to determine your insurance rates. They look at the conditional probability of you having accident, given your age, your car, your car color, your driving history, etc., and price your policy based on that likelihood.

View more about conditional probability in the following video.

### Example

If you draw two cards from a deck, what is the probability that you will get the Ace of Diamonds and a black card?

These two playing card scenarios are discussed further in the following video.

### Try It

### Example

A home pregnancy test was given to women, then pregnancy was verified through blood tests. The following table shows the home pregnancy test results.

Find

*P*(not pregnant | positive test result)*P*(positive test result | not pregnant)

Positive test | Negative test | Total | |

Pregnant | 70 | 4 | 74 |

Not Pregnant | 5 | 14 | 19 |

Total | 75 | 18 | 93 |

See more about this example here.