Independence and Conditional Probability

Day 2 and 3

More On Independence:•Two events are said to be ____________ if the

probability of the second is not effected by the first event happening.

• 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?

Proving Independence:

•Can use the Multiplication Rule to prove:

•So…

If then the two events are independent

•A and B are independent events such that P(A) = 0.5 and Find P(B).

( ) 0.7.c cP 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) Are having health insurance and a retirement plan independent events? Explain.• c) Are having these two benefits mutually exclusive? Explain.

Conditional Probability

•The probability that one thing happens given something else has happened.

•Said:

• 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?

Venn Diagrams and Probability

Venn diagrams can be used to illustrate the _______________ of a _________________.

In the first Venn Diagram, the complement of an event A, __________, contains exactly the outcomes that are not in A.

In the second Venn Diagram, A and B are ____________________ because they do not overlap (they have no outcomes in common).

Venn Diagrams and ProbabilityRecall the example on gender and pierced ears. We can use a Venn

diagram to display the information and determine probabilities.

Define events A: is male and B: has pierced ears.

In an apartment complex, 40% of residents read USA Today. Only 25% read the New York Times. Five percent of residents read both papers. Suppose we select a resident of the apartment complex at random and record which of the two papers the person reads.

a) Make a two-way table that displays the sample space of this chance process.

b) Construct a Venn diagram to represent the outcomes of this chance process.

c) Find the probability that the person reads at least one of the two papers. (Meaning: Reads one or the other or both)

d) Find the probability that the person doesn’t read either paper.

According to the National Center for Health Statistics, in December 2008, 78% of US households had a traditional landline telephone, 80% of households had cell phones, and 60% had both. Suppose we randomly selected a household in December 2008.

a) Make a two-way table that displays the sample space.

b) Construct a Venn Diagram.

c) Find P(a household has at least one of the two types of phones).

2. Two events A and B are such that P(A) =. 2 and

P(B) =.3 and P(A U B) = .4 Find the following probabilities. Use a Venn Diagram to represent each situation.

a) P(A and B) b) P(Ac U Bc)c) P(Ac and Bc)

3. A smoke detector system uses 2 devices, A and B. If smoke is present, the probability that it will be detected by device A is .95, by device B, .90 and by both devices .88.

a) If smoke is present, find probability that the smoke will be detected by either device A or B or both devices.

b) Find probability that smoke will be undetected.

The word problems…

•We can use the Multiplication Rule to do other forms of conditional probability problems…we need to rewrite the formula

•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.

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

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?