## 7.6 Conditional Probability and Independence

### Part A: Calculating Conditional Probability

(Based on Section 7.6 in Finite Mathematics and Finite Mathematics and Applied Calculus)

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Q What is conditional probability all about?
A Here is a quick illustration of conditional probability. Suppose you cast two dice; one red, and one green. Then the probability of getting "bulls eyes" (two ones) is 1/36. However, if, after casting the dice, you ascertain that the green die shows a one (but know nothing about the red die), then there is a 1/6 change that both of them will be one. In other words, the probability of "bulls eyes" changes if you have partial information, and we refer to this (altered) propbability as "conditional probability". (We will give you a more precise definition below.)

You might also want to take a look at our take a look at our on-line simulation of the "Monty Hall" game show (as discussed in the Chapter 7 You're the Expert Section in Finite Mathematics and Finite Mathematics and Applied Calculus ) now or after the tutorial to get more of a feel for conditional probability. Alternatively, read on...

First, a warmup on estimated probability to get started... (To review estimated probability, go back to Tutorial 7.2.)

Here is a table showing fictitious trial results of a new acne cream.
 Used Cream Used Placebo Skin Improved 800 600 No Improvement 400 200

The sample size (total number of people in the study) is:

.

The estimated probability that someone's skin improved (regardless of which skin cream was used) is:

.

The experimental probability that someone's skin improved, given that they used the new acne cream, is:

.

Now think about what these answers tell you about the acne cream's effectiveness. Does this seem at odds with the data in the table?

 Conditional Probability The probability that you just computed (last one above) The probability that someone's skin improved, given that they used the new acne cream, is an example of conditional probability, the probability of the event E, given the event F, and written P(E|F) (probability of E, given F)

Q How do we calculate conditional probability?
A Look at how we calculated the answer in the last question above. We used the ratio

P(E|F) =  number of subjects who used the cream and whose skin improvedtotal number of people who used the cream
P(E|F) =  Number of outcomes in E FTotal number of outcomes in F
P(E|F) =  fr(EF)fr(F)
P(E|F) =  P(EF)P(F)
We divided top and bottom by the total sample size

Calculating Conditional Probability

If E and F are events, then the probability of E given F is

 P(E|F) = P(EF)P(F)

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

 P(E|F) = n(EF)n(F)

(Recall that n(G) means the number of outcomes in the event G.)

For experimental probability, we can also use the alternative formula

 P(E|F) = fr(EF)fr(F)

(Recall that fr(G) means the frequency of the event G.)

Of Colossal Conglomerate's 16,000 clients, 3200 own their own business, 1600 are "gold class" customers, and 800 own their own business and are also "gold class" customers. What is the probability that a randomly chosen client who owns his or her own business is a "gold class" customer?

 0.25 0.50 0.10 0.05

You have invested in Home-Clone Inc. stocks, as you suspect that the company's "Clone-a-Sibling" kit will shortly be approved by the FDA. There is an 80% chance that FDA approval will be given, and a 95% chance that the value of the stock you hold will double if FDA approval is given. What is the probability that the FDA will approve the product and the value of the stock you hold will double?

 76% 84.2% 7.6% 77.5%

You now have several options:

• Go on to Part B by pressing the "Next Tutorial" button on the left
• Try some of the questions in the true/false quiz (warning: it covers the whole of chapter 76) by pressing "Chapter Quiz" on the left
• Try some of the exercises in Section 7.6 of Finite Mathematics, or Finite Mathematics and Applied Calculus.

Last Updated: July, 2000