7.5 Conditional Probability and Independence
This tutorial: Part A: Conditional Probability
Q: What is conditional probability all about?
A: Here is a quick illustration of what it means. Suppose you cast two dice; one red, and one green. Then let B be the event that you get "bulls eyes" (two ones). We know that

P(B) = \frac{1}{36}.

G: The green dice shows a one

P(B\G) = \frac{1}{6}

The probability of bulls eyes, given that the green die shows a 1, equals
\frac{1}{6}
Here is a table showing fictitious trial results of a new acne medication:
Total  
Total 
(This kind of table is sometimes referred to in statistics as a cross tabulation or contingency table.) Calculate the following:
 (Note: You can enter all answers as fractions.)
The probability that a participant's (regardless of treatment) is:

If we knew beforehand that the participant , then the probability that their is:

This second probability is a conditional probability: The probability that a participant's given that they If we write

E: A participant's
F: A participant
P(E\F)  =  
=  \frac{Number of outcomes in E \cap F}{Number of outcomes in F}  
=  \frac{n(E \cap F)}{n(F)}  
=  \frac{P(E \cap F)}{P(F)} Divide top and bottom by n(S) 
Conditional Probability
If E and F are events, then the probability of E given F is defined as
If all outcomes are equally likely, then we can also use the alternative formula
For relative frequency, we can also use

Examples
1. If there is a 10% chance that the moon will be in the Seventh House and Jupiter will also align with Mars, and a 25% chance that Jupiter will align with Mars, then what is the probability that the Moon is in the Seventh House given that Jupiter aligns with Mars? Here, take
J: Jupiter aligns with Mars P(M\J) = \frac{P(M \cap J)}{P(J)} = \frac{.10}{.25} = .4 3. Cast two dice (one red, and one green) and add the numbers facing up. Let
F: The sum is .

Conditional Probability of What given What?
Sometimes it is a little tricky to tell from the wording whether a statement refers to P(E\F) or P(F\E). The trick is to reword the statement in the form
 The probability that _______ given that (or assuming that) ________ is _________.
 20% of employees who took the course improved their productivity.
Reworded: The probability that an employee's productivity improved, assuming that they took the course, is .20. Therefore, P(I\T) = .20  80% of employees who improved their productivity had taken the course.
Reworded: The probability that an employee had taken the course, assuming that their productivity improved, is .80. Therefore, P(T\I) = .80  The probability that a tirerelated accident results in a rollover is .35.
Reworded: The probability that a [vehicle has a] rollover, given that it has had a tirerelated accident, is .35. Therefore, P(R\T) = .35  The probability that a rollover has resulted from a tirerelated accident is .65.
Reworded: The probability that a [vehicle has had a] tirerelated accident, given that it has had a rollover, is .65. Therefore, P(T\R) = .65
Here are some for you:
Select the appropriate formula for each statement:
Some of Colossal Conglomerate's clients are "gold class" clients, and others are "platinum class" clients. Here is a contingency table showing the number of clients in various categories:
Total  
Total 
Assume a Colossal Conglomerate client is chosen at random. Calculate the following:
 (Note: You can enter all answers as fractions.)
The probability that a client is is:

The probability that is a client is:

You have invested in HomeClone Inc. stocks, as the FDA is close to a decision as to whether to approve the company's "CloneaSibling" kit. There is a chance of FDA approval, a chance that the stock will double, and
You now have several options:
 Go on to Part B by pressing the "Next Tutorial" link on the left
 Try some of the questions in the true/false quiz (warning: it covers the whole of chapter 7) by going to "Everything for Finite Math"
 Try some of the exercises in Section 7.5 of or .
Copyright © 2009 Stefan Waner