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# Probability Bays theorem

Here are a set of problems that I would like to learn how to do the steps to. I have also prepared the questions with the answers, however would like to see the process in which the answers were derived.

(See attached file for full problem description)
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1. If P(A) = 0.5, P(B)= 0.4, and P(B&#9474;A) = 0.3, then P(A and B) is

2. If A and B are mutually exclusive events and P(A) = 0.3 and P(B) = 0.6, then P(A and B) is

3. How many ways can a Math Club schedule 4 speakers for 4 different meetings if they are all available on any of 6 possible dates?

4. A human gene carries a certain disease from mother to the child with a probability of 0.65. Suppose a mother of two children carries the gene. Also, assume that infection from child to child is independent. The probability that both children are infected with the disease is

5. A lot contains 20 fuses of which 5 are defective. If two fuses are selected at random without replacement, what is the probability that at most one is defective?

2 0 + 1 1
20
2

6. If A and B are two mutually exclusive events with P(A) = 0.15 and P(B) = 0.4, then P(A and B'), (i.e. probability of A and B complement) is

7. On a particular college campus, 65% of the non-traditional students are smokers. Research indicates that 15% of the smokers have some form of lung cancer. The probability of a non-traditional student on this campus having lung cancer given that the student is a smoker is

8. A large industrial firm uses three different warehouses (A, B and C) to store its manufactured product. From past records, it is known that 20% of the manufactured product are assigned to warehouse A, 50% are assigned to B, and 30% to warehouse C. If it is known that 5% of the product in warehouse A are defective, 4% in warehouse B are defective, and 8% in warehouse C are defective, what is the probability that if a product is selected at random from one of these warehouses that it will be defective?

9. A large industrial firm uses three different warehouses (A, B and C) to store its manufactured product. From past records, it is known that 20% of the manufactured product are assigned to warehouse A, 50% are assigned to B, and 30% to warehouse C. If it is known that 5% of the product in warehouse A are defective, 4% in warehouse B are defective, and 8% in warehouse C are defective, what is the probability that if a defective product is selected at random that it came from warehouse C?

10. From a group of 5 men and 6 women, how many committees of size 3 are possible with two men and 1 women if a certain man must be on the committee?

Answer: 4 x 1 x 6
1 1 1

11. A shipment of 12 calculators contains 3 defective calculators. In how many ways can a school purchase 5 of these calculators and receive at least two of the defective calculators?

Answer: 3 x 9 + 3 x 9
2 3 3 2

12. The probability that a patient recovers from an operation is 0.9. What is the probability that exactly 2 of the next 3 patients who have this operation will recover?