Chapter 7: Discrete Probability

Question 2. To determine

1. What probability distribution for birthday’s should be used to reflect how often February$\mathbf{29}$occurs?
2. What is the probability that in a group of$\mathit{n}$people at least two have the same birthday using the probability distribution from$\mathit{a}$?

Question: What is the probability that a positive integer not exceeding $\mathbf{100}$selected at random is divisible by $\mathbf{3}$?

Question: Suppose that\(n\)balls are tossed into\(b\)bins so that each ball is equally likely to fall into any of the bins and that the tosses are independent.

a) Find the probability that a particular ball lands in a specified bin.

b) What is the expected number of balls that land in a particular bin?

c) What is the expected number of balls tossed until a particular bin contains a ball?

d) What is the expected number of balls tossed until all bins contain a ball? (Hint: Let\({X_i}\)denote the number of tosses required to have a ball land in an\(ith\)bin once\(i - 1\)bins contain a ball. Find\(E\left( {{X_i}} \right)\)and use the linearity of expectations.)

Question 23. To determine

What is the conditional probability that exactly four heads appear when a fair coin is flipped five times, given that the first flip came up heads?

Question: Suppose that and are the events that an incoming mail \({E_1}\)message contains the words \({w_1}\) and \({w_2}\), respectively. Assuming that \({E_1}\) and \({E_2}\) are independent events and that \({E_1}\left| S \right.\) and \({E_2}\left| S \right.\) are independent events, where S is the event that an incoming message is spam, and that we have no prior knowledge regarding whether or not the message is spam, show that

\(p(S|{E_1} \cap {E_2}) = \frac{{p({E_1}|S)p({E_2}|S)}}{{p({E_1}|S)p({E_2}|S) + p({E_1}|\bar S)p({E_2}|\bar S)}}\)

Question: What is the probability that a positive integer not exceeding $\mathbf{100}$selected at random is divisible by $\mathbf{5}\mathbf{}\mathbf{\text{or}}\mathbf{}\mathbf{7}$?

Question: Use the law of total expectations to find the average weight of breeding elephant seal, given that 12% of the breeding elephant seals are male and the rest are female, and the expected weight of the breeding elephant seal is 4,200 pounds for a male and 1,100 pounds for a female.

Question: Suppose that \(A\) and \(B\) are events with probabilities \(p(A) = 3/4\) and \(p(B) = 3/4\).

a) What is the largest \(p(A \cap B)\) can be? What is the smallest it can be? Give examples to show that both extremes for \(p(A \cap B)\) are possible.

b) What is the largest \(p(A \cup B)\) can be? What is the smallest it can be? Give examples to show that both extremes for \(p(A \cup B)\) are possible.

Question: Let A be an event. Then IA, the indicator random variable of A, equals 1 if A occurs and equals 0 otherwise. Show that the expectation of the indicator random variable of A equals the probability of A, that is, E(IA)=p(A).

Question: Find the probability of winning a lottery by selecting the correct six integers, where the order in which these integers are selected does not matter, from the positive integers not exceeding.

(a) 30.

(b) 36.

(c) 42.

(d) 48.