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

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 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: We say: (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:


(This kind of table is sometimes referred to in statistics as a cross tabulation or contingency table.) Calculate the following:

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

then the probability we just computed is P(E\|F). We computed it by taking the ratio
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.


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


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

Here are some examples (the underlined letters give the names of the associated events.)

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:


Assume a Colossal Conglomerate client is chosen at random. Calculate the following:

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:

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Last Updated: June, 2009
Copyright © 2009 Stefan Waner