Question

1. Roll a pair of dice (one is red and the other is green). Let A be the event that the red die is 4 or 5. Let B be the event that the green die is 1. Let C be the event that the dice sum is 7 or 8.

1. Calculate P(A), P(B), P(C)
2. Calculate P(A|C), P(A|B)
3. Are the events A and C independent?

                                                                                                                                    (20 points)

2. Suppose box 1 has four black marbles and two white marbles, and box 2 has two black marbles and five marbles. If you picked one marble from one of the two boxes at random, what it the probability that you picked from box 1 given that the marble you picked is black?

(10 points)

3. A raffle has 5000 tickets with the following prizes: 1 ticket has $2000 prize, 10 tickets have $200 prize, and 20 tickets have $50 prize and 500 tickets have a $20 prize. If to buy a ticket costs $15, and X is the random variable that measures net profit:

1. Calculate the pdf table of X
2. Calculate E(X), Var(X)

                                                                                                                                    (20 points)

4. If a fair coin is flipped 120 times, what is the probability that:

1. The number of heads is more than 70
2. The number of heads between 50 and 70?

(20 points)

5. According to a study, 21.1% of 507 female college students were on a diet at the time of the study.

a) Construct a 99% confidence interval for the true proportion of all female students who were on a diet at the time of this study.

b) Explain what this interval means.

c) Is it reasonable to think that only 17% of college women are on a diet?

                                                                                                                                   

                                                                                                                                    (20 points)

6. A used car dealer says that the mean price of a 2008 Honda CR-V is at least $20,500. You suspect this claim is incorrect and find that a random sample of 14 similar vehicles has a mean price of $19,850 and a standard deviation of $1084. Is there enough evidence to reject the dealer’s claim at α = 0.05?

                                                                                                                                    (10 points)

Answer 1

A) P(A) = 2/6

There are two ways of getting 4 or 5 on red die out of 6

P(B) = 1/6

There is one ways of getting 1 on green die

P(C) = Probability of getting sum 7 + probability of getting sum 8

= 6/36 + 5/36

=11/36

b. P(A/C) = 4/36

Event A/C is getting 4 or 5 on red die given sum of die is 7 or 8

P(A/B).= P(A) = 2/6

Since number that turn up.on red die is independent of number of green die, event A and B are independent, Hence P(A/B) = P(A)

C. Event A and C are not independent as number on red die effect sum of two die.