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}.

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
Copyright © 2009 Stefan Waner