# Calculating Conditional and Unconditional Probabilities

The probability of having disease X in the general population is only .05. The Sagman Test is a newly discovered method for early detection. Of those who have disease X, the test indicates the disease for 90% of them. Of those who do not have the disease, the test indicates no disease for 90% of them. Is the test a good predictor of whether you actually have the disease or not? Should its use be widespread?

a. Which of the following conditional probabilities is most appropriate to determine the test's effectiveness? Explain.

P(have disease X / test says you have the disease)

P(test says you have the disease / have the disease)

P(test says you have the disease)

b. Compare the unconditional probability of having the disease to the conditional probability of having the disease given the test says you have the disease.

c. Compare the unconditional probability of not having the disease to the conditional probability not having the disease given the test says you do not have the disease.

d. Based on parts b and c, is the test a good predictor? Explain.

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#### Solution Preview

The probability of having disease X in the general population is only .05. The Sagman Test is a newly discovered method for early detection. Of those who have disease X, the test indicates the disease for 90% of them. Of those who do not have the disease, the test indicates no disease for 90% of them. Is the test a good predictor of whether you actually have the disease or not? Should its use be widespread?

Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read "the (conditional) probability of A, given B" or "the probability of A under the condition B". When in a random experiment the event B is known to have occurred, the possible outcomes of the experiment are reduced to B, and hence the probability of the occurrence of A is changed from the unconditional probability into the conditional probability given B.

The conditional probability can be calculated as P(A | B) = P(A and B)/ P(B)

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a. Which of the following conditional probabilities is most appropriate to determine the test's effectiveness? Explain.

P(have disease X / test says you have the disease) ...

#### Solution Summary

The solution helps with questions that involve conditional and unconditional probability calculations.

Fault Tree Problem: Calculating probabilities.

Assume the probability of a tire blowout is 0.178% per 50,000 miles of use and that a person travels 15,000 miles per year in a car. Assume that the probability of loss of control is 60% if the blowout occurs on the front tires and 20% for the rear tires. If control is lost, the probability is 50% of veering to the right and 50% of veering to the left. If it goes left, the car spends 1 sec crossing the one lane of oncoming traffic (2000 vehicles/hour). Either way it goes, it ends up on the roadside, with a 20% chance of hitting a barrier or tree, if not hit by oncoming traffic. Assume there is a 20% chance of death if the car hits a barrier or tree and a 50% chance of death if hit by oncoming traffic. Calculate the probability of being killed per year in this type of accident

Hint: This is a fault tree problem. Practice with a simpler example from the readings/notes. When working the real problem, sum up all the ways of having a fatal accident and their probabilities.

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