Math 170E

Practice for Midterm

Summer 2024

Problem 1. We have two fair, six-sided dice, one of which is red and the other of which is blue. We roll both dice.

(a) What is the probability that at least one of the dice lands on an odd number?

(b) What is the probability that the result of the blue die is strictly less than the result of the red die?

(c) Let A be the event that the sum of the rolls is even. Let B be the event that the results of both rolls are numbers ≥ 4. Are A and B independent events? Justify your answer.

Problem 2. We have a bag with 7 green chips and 11 yellow chips.

(a) You draw from the bag 6 times with replacement. Find the probability that the number of yellow chips drawn during the first three draws is exactly the same as the number of yellow chips drawn during the last three draws.

(b) You draw from the bag 6 times with replacement. Let X be the number of green chips drawn this way. Given that 2 out of the first 3 draws were green chips, what is the conditional probability that X = 4?

(c) Suppose now you draw 3 chips from the bag without replacement. Let Y be the number of green chips drawn this way. Given that Y = 2, what is the conditional probability that the first two draws were both green chips?

Problem 3. A publisher offers an unusual deal to an unknown author. The publisher will take the entirety of the first $10, 000 in profit of the book sales; all profit made after this initial $10, 000 in profit will go to the author. (For example, if the book makes $14, 000 in total profit, the publisher will earn $10, 000 while the author will earn $4, 000.)

Let X be the total profit made from the book sales (before it is split between the publisher and the author). Assume that the pmf of X is

Let Y be the amount of the profit that will go to the author.

Compute E[Y ] and Var(Y ).

Problem 4. You are dealt (without replacement) 6 cards from a standard deck of 52 playing cards.

(a) What is the probability that in your hand of 6 cards you have exactly two pairs?

(b) Suppose now that you are dealt one more card to add to your hand. Given that your initial hand of 6 cards had exactly two pairs, what is the conditional probability that your hand of 7 cards has exactly three pairs?

Problem 5. Arrivals come according to a Poisson process with rate λ = 3. The process starts at time t = 0.

(a) Let N be the number of arrivals in the interval of time [2, 4]. Compute the conditional probability P(N ≤ 3|N ≥ 1).

(b) Let T10 be the time of the time of the tenth arrival. Find numbers a and b such that

(c) Let T3 be the time of the third arrival. Compute E[1/X2].

Problem 6. Let X have chi square distribution with 1 degree of freedom, X ~ χ2(1). Define a new random variable as follows: sample a value of X and then flip a fair coin. If the coin lands heads, then Y = p X. If the coin lands tails, then Y = − p X. Show that Y is a standard normal random variable. Hint. Start with computing P(Y ≤ y) in two cases: y ≥ 0 and y < 0.

Problem 7. You and a friend are meeting at an agreed upon location. You will arrive at time t = 1, whereas your friend’s arrival time is not so certain. We model your friend’s arrival time with a continuous random variable which is uniform. on [0, 2].

Let W be the amount of time you wait for your friend once you arrive. Note that if your friend arrives before or at time t = 1, then W = 0.

(a) Find the cdf of W.

(b) Is W discrete, continuous, or neither? Justify your answer.