## 7.5 Conditional Probability and Independence if (parent.playingGame) document.writeln('<i><font color = AA00DD>Game Version</font></i>')

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}.
Suppose however that, after casting the dice, you find out that the green die shows a one (but you still know nothing about the red die). Then there is now a 1/6 chance that both of them will be one, since the probability that the red die is one is 1/6. In other words, the probability of B has changed once you know that the related event
G: The green dice shows a one
has occurred. We call this (altered) probability of B the conditional probability of bulls eyes, given that the green die shows a one; that is, the conditional probability of B given G. We write:
P(B\|G) = \frac{1}{6}
We say:
The probability of bulls eyes, given that the green die shows a 1, equals \frac{1}{6}
(There is a more precise definition below.) The following quiz will lead us to the mathematical definition of conditional probability:

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:

 The probability that a participant's (regardless of treatment) is:         (Note: You can enter all answers as fractions.) 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
then the probability we just computed is P(E\|F). We computed it by taking the ratio
 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

P(E\|F) = \frac{P(E \cap F)}{P(F)}

If all outcomes are equally likely, then we can also use the alternative formula

P(E\|F) = \frac{n(E \cap F)}{n(F)}

For relative frequency, we can also use

P(E\|F) = \frac{fr(E \cap F)}{fr(F)}       Recall that fr(G) means the frequency of the event G.

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

M: The Moon is in the Seventh House
J: Jupiter aligns with Mars

P(M\|J) = \frac{P(M \cap J)}{P(J)} = \frac{.10}{.25} = .4

2.

3. Cast two dice (one red, and one green) and add the numbers facing up. Let

E: The green die shows .
F: The sum is .
 P(E\|F) =

 P(F\|E) =

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 _________.
Here are some examples (the underlined letters give the names of the associated events.)
• 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 tire-related accident results in a rollover is .35.
Reworded: The probability that a [vehicle has a] rollover, given that it has had a tire-related accident, is .35. Therefore, P(R\|T) = .35
• The probability that a rollover has resulted from a tire-related accident is .65.
Reworded: The probability that a [vehicle has had a] tire-related 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:

 The probability that a client is is:         (Note: You can enter all answers as fractions.) The probability that is a client is:

You have invested in Home-Clone Inc. stocks, as the FDA is close to a decision as to whether to approve the company's "Clone-a-Sibling" kit. There is a chance of FDA approval, a chance that the stock will double, and

The probability that the FDA will approve the product and the value of the stock you hold will double is:

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 .

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Last Updated: June, 2009