Section 6.5: Conditional Probability and Independence

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Part A. Calculating Conditional Probability

Part B. Trees and Conditional Probability

Part C. Independent Events

Part A: Calculating Conditional Probability

A warmup on experimental probability to get started... To review experimental probability, go back to Tutorial 6.2 by either pressing here or pressing the "back" button on the sidebar twice

Here is a table showing a fictitious statistical study 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) was

	N = 1,200			N = 800			N = 2,000
	N = 1,400

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

	0.6			0.7			2/3
	1.0

Here is the table again (in case it's scrolled out of range).

	Used Cream	Used Placebo
Skin Improved	800	600
No Improvement	400	200

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

	0.6			0.7			2/3
	1.0

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?

Question
How do we calculate conditional probability?

Answer
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 improved total number of people who uised the cream
P(E|F)	=	number of outcomes in E F total number of outcomes in F
P(E|F)	=	fr(EF) fr(F)
P(E|F)	=	P(EF) P(F)	(dividing 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.)

The following quiz items are based on Examples 2 and 3 on pp. 444-445 of Finite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World.

Of Colossal Conglomerate's 16,000 clients, 3,200 own their own business, 1,600 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 "B" button below
• Try some of the questions in the true/false quiz (warning: it covers the whole of chapter 6) by pressing the button on the sidebar
• Try some of the exercises on pp. 453-457 of Finite Mathematics Applied to the Real World, or Finite Mathematics and Calculus Applied to the Real World. Exercises 1 through 44 require only the material from Part A.

Part B. Trees and Conditional Probability

Part C. Independent Events