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# Probability and the Bayes Theorem

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1. The information below is the number of daily emergency service calls made by the volunteer ambulance service of Walterboro, South Carolina, for the last 50 days. To explain there were 22 days on which there were 2 emergency calls, and 9 days on which there were 3 emergency calls.

Number of calls Frequency
0 8
1 10
2 22
3 9
4 1
Total 50

A. Convert this information on the number of calls to a probability distribution.
B. Is this an example of a discrete or continuous probability distribution?
C. What is the mean number of emergency calls per day?
D. What is the standard deviation of the number of calls made daily?

2. P(A1) = .60, P(A2) = .40, P(B1/A1) = .05, and P(B1/A2) = .10. Use Bayes' theorem to determine P(A1/B1).

3. Refer to the following table.

Second Event A1 A2 A3 Total
B1 2 1 3 6
B2 1 2 1 4

Total 3 3 4 10

A. Determine P(A1).
B. Determine P(B1/A2).
C. Determine P(B2 and A3).