MATH 230

TEST II

April 9, 1999

Bellamy & Sullivan

Problems 1 through 5 are worth 20 points each; problem 6 is worth 10 quiz points.

1. 20 points

   Let A and B be events in a sample space S with Pr(A)=3/8 and Pr(B)=2/3.

   1. What is the smallest possible value for Pr(A B)?
   2. Compute Pr(A B) if Pr(A B) has the smallest possible value.
   3. Compute Pr(A B) if A and B are independent.
   4. If C is another event in S and A and C are disjoint, what is the largest possible value for Pr(C). Could A and B be disjoint?

2. 20 points

   A fair die is rolled four times. Find, as a fraction in lowest terms, the probability that no number is rolled twice.
   

3. 20 points

   Twenty indepent Bernoulli trials are performed. The probability of success in each trial is 5/9. Compute (as fractions with denominator equal to 9) the expected value and standard deviation of the number of successes. Then write down, but do not simplify, the probability that exactly eleven successes are obtained.
   

4. 20 points

   (Do not simplify answers) Four members of a fifteen member engineering research development team are chosen to show the prototype of a new product at a trade show. There are eleven men and four women on the team, and the four members are chosen randomly.
   
   1. Find the probability that the four people chosen are all male.
   2. Find the conditional probability that all are male given that they are all the same sex.
   3. Find the conditional probability that the four chosen are all female given that at least three are female.
   
   

5. 20 points

   (Simplify answers) An urn contains three red balls and four green balls. A ball is chosen randomly, its color is noted, and it is put back in the urn. Then, seven more balls of the same color as the one drawn are added to the urn, and another ball is drawn randomly from the 14 now in the urn. Find:
   1. The probability that both balls drawn were red.
   2. The probability that both balls are green.
   3. The probability that both balls drawns are the same color.
   4. The probability that the first ball was green given that the second was green.
   
   

6. 10 Quiz Points

   Referring the problem 5, let X=the number of red balls drawn. Note that your answer to part a) of problem 5 is just Pr(X=2), while your answer to part b) is Pr(X=0). What other possible value could X have? Set up a probability table for X and compute E(X) as a fraction in lowest terms.