Alice and Bob arrange to meet for lunch on a certain day at noon. However, neither is

known for punctuality. They both arrive independently at uniformly distributed times

between noon and 1 pm on that day. Each is willing to wait up to 15 minutes for the

other to show up. What is the probability they will meet for lunch that day?

2. Alice, Bob, and Carl arrange to meet for lunch on a certain day. They arrive independently at uniformly distributed times between 1 pm and 1:30 pm on that day.

(a) What is the probability that Carl arrives first?

For the rest of this problem, assume that Carl arrives first at 1:10 pm, and condition on

this fact.

(b) What is the probability that Carl will have to wait more than 10 minutes for one of

the others to show up? (So consider Carl's waiting time until at least one of the others

has arrived.)

(c) What is the probability that Carl will have to wait more than 10 minutes for both

of the others to show up? (So consider Carl's waiting time until both of the others has

arrived.)

(d) What is the probability that the person who arrives second will have to wait more

than 5 minutes for the third person to show up?

3. One of two doctors, Dr. Hibbert and Dr. Nick, is called upon to perform a series of

n surgeries. Let H be the indicator r.v. for Dr. Hibbert performing the surgeries, and

suppose that E(H) = p. Given that Dr. Hibbert is performing the surgeries, each surgery

is successful with probability a, independently. Given that Dr. Nick is performing the

surgeries, each surgery is successful with probability b, independently. Let X be the

number of successful surgeries.

(a) Find the joint PMF of H and X.

(b) Find the marginal PMF of X.

(c) Find the conditional PMF of H given X = k.

4. A fair coin is flipped twice. Let X be the number of Heads in the two tosses, and Y be

the indicator r.v for the tosses landing the same way.

(a) Find the joint PMF of X and Y .

(b) Find the marginal PMFs of X and Y .

(c) Are X and Y independent?

(d) Find the conditional PMFs of Y given X = x and of X given Y = y

Part A. For two independent events A and B, suppose that P(A) = 0.2 and P(B) = 0.4. To find P(A /7 B), what probability rule/law, should you use? O Axioms of probability O Definition of the independent events O The law of total probability O The multiplication rule for P(A / B) O Definition of the conditional probability O Bayes' theorem O The addition rule Find P(A /) B). To find P(A U B), what probability rule/law, should you use? O The multiplication rule for P(A / B) O Definition of the conditional probability O Bayes' theorem O The law of total probability O The addition rule O Definition of the independent events O Axioms of probability Find P(A U B).2020 Goldman Sachs Engineering -- Combined 9 02 : 04 to test end 3/12 Attempted L yuyuarizhao Probability and Statistics - Simple Probability Let X have the probability density function given by: fx(x) = 0.5*e'll where -co