Conditional Probability

Conditional probability is a way of describing the influence of one event on another. Because much of statistics is about making predictions, conditional probability is one of the most important concepts in this course.

The statement, "Bernie Williams is a better hitter with runners in scoring position" is a statement about conditional probability. It says that likelihood of the event "Bernie Williams gets a hit" is positively related to the event "Bernie Williams comes to bat with runners in scoring position."

The statement, "The Democratic candidate is more likely to win if there is a higher voter turnout" is a statement about conditional probability. It says that the likelihood of the event "The Democratic candidate wins" is positively related to the event "Voter turnout is high."

Suppose that we are playing a game where we flip three coins with the object of having two or more flips come up heads. Before we start the game, the probability of winning is 4 out of 8, or 1/2.

After we flip one coin, our probability of winning the game changes. If the first coin comes up H, then our probability of winning the game is 3/4. We say that the conditional probability of the event "winning the game" given the event "the first coin comes up H" is 3/4. What is the conditional probability of winning the game given that the first coin comes up T?

In general, we define the conditional probability of event B given event A as

P(B|A) = P(A and B)/P(A)

That is, the conditional probability of event B given event A is equal to the probability of the compound event (A and B) divided by the probability of event A.

For example, consider the compound event, "Bernie Williams gets a hit with runners in scoring position." We might represent this in terms of A and B as follows:

event | description | probability |
---|---|---|

A | Bernie Williams comes up with a runner in scoring position | 0.2 |

B | Bernie Williams gets a hit | 0.3 |

A and B | Bernie Williams gets a hit with a runner in scoring position | 0.08 |

P(B|A) | Conditional probability of Bernie Williams getting a hit with a runner in scoring position | 0.4 |

Note that according to these data, P(B|A), the conditional probability of Bernie Williams getting a hit with a runner in scoring position, is higher than the unconditional probability of Bernie Williams getting a hit. Bernie Williams is a .400 hitter with a runner in scoring position, compared with a .300 hitter in general.

What is the probability of the event (A and B^{c}), of Bernie Williams coming up with a runner in scoring position and not getting a hit? We know that

P(A and B) + P(A and B^{c}) = P(A)

Therefore, P(A and B^{c}) = 0.2 - 0.08 = .12. Fill out the rest of the table below.

event | description | probability |
---|---|---|

A and B^{c} |
Bernie Williams comes up with a runner in scoring position and does not get a hit | 0.12 |

B and A^{c} |
Bernie Williams comes up without a runner in scoring position and gets a hit | ? |

A^{c} and B^{c} |
Bernie Williams comes up without a runner in scoring position and does not get a hit | ? |

P(B|A^{c}) |
Conditional probability of Bernie Williams getting a hit without a runner in scoring position | ? |

We know that Bernie Williams is a .400 hitter with runners in scoring position. What is his batting average without a runner in scoring position? If you had no idea whether or not a runner is in scoring position, what would be your best estimate of the probability of Bernie Williams getting a hit? How does knowing whether or not a runner is in scoring position improve the accuracy of your prediction?