INDEPENDENCE AND

CONDITIONAL PROBABILITY

THE MULTIPLICATION RULE (INTERSECTION) Symbol: - What they have in

common

Events A and B are independent if: Knowing that one occurs does not

change the probability that the other occurs.

Rule: P(A and B) =

AND means to multiply!!

( ) ( ) ( )P A B P A P B

Events A and B are dependent if: One event does change the probability

of the next event.

The Rule: P(A and B)=

In other words: Probability of event A occurs multiplied

by the probabilty of event B occurs given event A occurs.

THE MULTIPLICATION RULE (INTERSECTION)

( ) ( ) ( | )P A B P A P B A

IMPORTANT DISTINCTION: Disjoint or mutually exclusive CAN NOT

be independent!!

Since they have no common outcomes, knowing that one occurs means the other didn’t.

The are DEPENDENT

EXAMPLES A general can plan a campaign to fight

one major battle or 3 small battles. He believes he has a probability of .6 or winning the large battle and a probability of .8 of winning each small battle. Victories or defeats in small battles are independent. The general must win either the large battle or all three of the small battles. Which strategy should he choose?

INDEPENDENCE AGAIN… Two events are said to be independent if the

probability of the second is not effected by the first event happening.-The results do no effect each other.

Independent or Dependent? Calling in to a radio station and winning their

radio contest? Selecting an ace from a deck returning it and

then selecting another ace from the deck? Rolling a twelve on a pair of dice, then rolling

a twelve on the same pair of dice?

Can use the Multiplication Rule to prove: So… If then the two

events are independent( ) ( ) ( )P A B P A P B

A and B are independent events such that

P(A) = 0.5 and Find P(B).( ) 0.7.c cP A B

( ) ( ) ( ) ( )c c c c c cP A B P A P B P A B

For events C and D, P(C) = 0.7 and P(D) = .3 and P(C U D) = 0.9. Find Are C and D independent? Why or why not?

( ).P C D

Most sample surveys use random digit dialing equipment to call residential telephone numbers at random. The telephone polling firm Zogby International reports that the probability that a call reaches a live person is 0.2. Calls are independent.

a) A polling firm places 5 calls. What is the probability that none of them reaches a person.

b) When calls are made to NYC, the probability of reaching a person is only 0.08. What is the probability that none of 5 calls made to NYC reaches a person.

Fifty-six percent of all American workers have a workplace retirement plan, 68% have health insurance, and 49% have both benefits. We select a worker at random.

a) What’s the probability he has neither employer-sponsored health insurance nor a retirement plan?

b) What’s the probability he has health insurance if he has a retirement plan?

c) Are having health insurance and a retirement plan independent events? Explain.

d) Are having these two benefits mutually exclusive? Explain.

CONDITIONAL PROBABILITY The probability that one thing happens

given something else has happened.

Probability of A given B has happened.

( / )P A B

In an effort to reduce the amount of smoking, administration of Podunk U is considering establishing a smoking clinic to help students. A survey of 1000 student a the school was conducted and the results are given below:

Voted against the clinic? Voted against the clinic given that he is a frosh? Is a frosh given that he voted against the clinic? Is a junior given that he didn’t vote against the

clinic?

THE WORD PROBLEMS… We can use the Multiplication Rule to do

other forms of conditional probability problems…we need to rewrite the formula

EXAMPLES: In Ashville the probability that a married

man drive is 0.9. If the probability that a married man and his wife both drive is 0.85, what is the probability that the wife drives given that he drives?

In a certain community the probability that a man over 40 is overweight is 0.42. The probability that his blood pressure is high given that he is overweight is 0.67. If a man over 40 is selected at random, what is the probability he is overweight and has high blood pressure?

Dr. Carey has 2 bottles of sample pills on his deck for treatment of arthritis. One day he gives Mary a few pills from one of the bottles but he doesn’t remember which bottle he took the pills from. The pills in bottle A are effective 70% of the time with no known side effects. The pills in bottle B are effective 90% of the time with some side effects. Bottle A is closer to Dr, Carey and he has a probability of 2/3 that he selected from this one.

A) Find the probability that the pills are effective.

B) What is the probability that the pills came from Bottle A given that they are effective.

A test for a certain disease has the following properties: the test is positive 98% of the time for persons who have the disease. The test is also positive for 1% of the time for those who don’t have the disease. Studies have established that 7% of the population has the disease.

What is the probability that a person chosen at random will test positive?

What is the probability that a person who tests positive actually has the disease?

What is the probability of a false negative?