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?

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?

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)

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?

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?

                                                                                                                                       

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?

Answer 1

a) P(A) = Probability that the red die comes out to be a 4 or a 5

= 2/6 = 1/3

Therfore P(A) = 1/3 is the required probability here.

P(B) = Probability that the green die is a 1.
= 1/6

Therefore P(B) = 1/6 is the required probability here.

P(C) = Probability that the dice sum is 7 or 8:
Sum of 7 is obtained as: 1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2 or 6 + 1 that is 6 cases.
Sum of 8 is obtained as: 2 + 6, 3 + 5, 4 + 4, 5 + 3 or 6 + 2 that is 5 cases.

Total cases: 6*6 = 36

Therefore the probability now is computed here as:

= (6 + 5)/36

= 11/36

Therefore 11/36 is the required probability here.

b) The probability here are computed using Bayes theorem as:

P(A | C) = P(A and C) / P(C)

= Prob. that red die is 4 or 5 and the sum is a 7 or an 8 / P(C)

4 + 3, 4 + 4, 5 + 2, or 5 + 3 are the 4 cases here.

Therefore the probability here now is computed as:

= 4 / 11

Therefore 4/11 is the required probability here.

P(A | B) : Given that green die gets a 1, probability that there is a 4 or a 5 on blue die is computed here as:

= P(A) because A and B are independent events. Whatever happens on green die wont have an effect on the outcome of blue die.

= 1/3

Therefore 1/3 is the required probability here.