10 - Tamalpais Union High School District



Dependent Events and Conditional Probability

Introduction: Medical tests can be used for many purposes: to determine whether a person has a disease, (say, AIDS), carries a disease-causing organism (say, HIV), has used certain drugs to enhance athletic performance, or has used an illegal recreational psychoactive substance. But these tests are sometimes wrong. They can be wrong in either of two ways: They can turn out negative when they should be positive, or they can turn out positive when they should be negative. If we are testing for drug use and the individual has not used drugs, a mistaken positive result is said to be a “false positive.” If the individual has used drugs but the test result is negative, it is said to be a “false negative.”

|Example 1: The ELISA test is used to screen donated blood for the presence of | |

|HIV antibodies. When HIV antibodies are present in the tested blood, ELISA | |

|gives a positive result 98% of the time. When HIV antibodies are not present | |

|in the blood tested, ELISA gives a positive result 7% of the time (a false | |

|positive.) Suppose that 1 out of every 1000 units of blood donated actually | |

|contains HIV antibodies. | |

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|What is the probability that a positive ELISA result for a unit of donated | |

|blood is accurate? | |

|If 1 out of every 1000 units of donated blood is contaminated, what is the | |

|probability that a negative ELISA result is accurate? | |

|Which do you feel is more important- an accurate negative result or an | |

|accurate positive result? Why? | |

|Example 2: For men, binge drinking is defined as having five or more drinks in| |

|a row, and for women as having four or more drinks in a row. According to a | |

|study by the Harvard School of Public Health, 44% of college students engage | |

|in binge drinking, 37% drink moderately, and 19% abstain entirely. Another | |

|study by the American Journal of Health Behavior found that among college aged| |

|binge drinkers, 17% have been involved in an alcohol related automobile | |

|accident, while among non-binge drinkers, only 9% have been involved in such | |

|accidents. | |

| | |

|What’s the probability that a student who is selected at random has had an | |

|alcohol-related car accident? | |

|What’s the chance that a student who has an alcohol related car accident is a | |

|binge drinker? | |

|Problem A. Police checkpoints often set up sobriety checkpoints- roadblocks where drivers are asked a few brief questions to allow the officer to judge |

|whether or not the person may have been drinking. If the officer does not suspect a problem, drivers are released to go on their way. Otherwise, drivers are |

|detained for a Breathalyzer test that will determine whether or not they are arrested |

|The police say that based on a the brief initial stop, trained officers can |a. You are stopped at the checkpoint and, of course, you have not been |

|make the right decision 80% of the time. Suppose the police operate a sobriety|drinking. What’s the probability that you are detained for further testing? |

|checkpoint after 9pm on a Saturday night, a time when national traffic safety | |

|experts suspect that about 12% of drivers have been drinking. | |

| |b. What’s the probability that any given driver will be detained? |

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| |c. What’s the probability that a driver who is detained has actually been |

| |drinking? |

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| |d. What’s the probability that a driver who was released has actually been |

| |drinking? |

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|Problem C. A patient with a failing heart has been told that the probability of|a. Find the probability that the patient survives. |

|finding a suitable transplant is 0.38, the conditional probability of surviving| |

|GIVEN that a transplant is performed is 0.85, and the conditional probability | |

|of surviving GIVEN that a transplant operation is not performed is 0.30. |b. Find the probability that a suitable transplant was found and the patient |

| |survived. |

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| |c. Find the conditional probability that a transplant was found GIVEN that the |

| |patient survived. |

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REVIEW

1. Real estate ads suggest that 64% of homes for sale have garages, 21% have swimming pools, and 17% have both.

A. Draw a Venn Diagram. B. What’s the probability that a home for sale has a pool or a garage, but not both?

C. Neither a pool nor a garage? D. A pool, but no garage?

2. In its monthly report, a local animal shelter states that it currently has 24 dogs and 18 cats available for adoption. Eight of the dogs and 6 of the cats are male. Find each of the following conditional probabilities if an animal is selected at random:

A. the pet is male, given that it is a cat. B. The pet is a cat, given that it is female.

C. the pet is female, given that it is a dog. (hint- you may wish to make a 2-way table.)

| |Blood Pressure |

|Cholesterol | |High |OK |

| |High |0.11 |0.21 |

| |OK |0.16 |0.52 |

3. The probabilities that an adult American man has high blood pressure and/or high cholesterol are given in the table.

A. What’s the probability that a man has both conditions?

B. what’s the probability that he has high blood pressure?

C. what’s the probability that a man with high blood pressure has high cholesterol?

D. what’s the probability that a man has high blood pressure if it’s known that he has high cholesterol?

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