# Probability and the Bayes Theorem

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).

https://brainmass.com/statistics/bayesian-probability/probability-and-byaes-theorem-240732

#### Solution Summary

This solution practices the application of the Bayes theorem.

The probability

A young engineer has invented holographic mobile phones and has approached a venture capital company to invest in it. The venture capital company considers the product to be an all or nothing product: either everyone will want one because everyone else has one or no one will want one because there will be no one to use it with. The company believes that the probability that it will take off netting them a profit of $2000000 is 0.14. If it doesn't take off then they expect that they would loose $200000. They are considering using a consumer survey to gather more information. However, the company has experience that shows that the probability that the consumer survey will predict success for a product that will fail is 0.24, and the probability that the consumer survey will predict failure when the product will be a success is 0.07. What is the monetary value of the information from a consumer survey to the venture capital company in this case? (ie what is the maximum that they should spend on a consumer survey)?

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