Suppose that A, B, and C are three independent events such

If A and B are independent events and Pr(B) < 1, what is the value of Pr(Ac|Bc)?

Assuming that A and B are independent events, prove that the events Ac and Bc are also independent.

Suppose that A is an event such that Pr(A) = 0 and that B is any other event. Prove that A and B are independent events.

Suppose that a person rolls two balanced dice three times in succession. Determine the probability that on each of the three rolls, the sum of the two numbers that appear will be 7.

Suppose that the probability that the control system used in a spaceship will malfunction on a given flight is 0.001. Suppose further that a duplicate, but completely independent, control system is also installed in the spaceship to take control in case the first system malfunctions. Determine the probability that the spaceship will be under the control of either the original system or the duplicate system on a given flight.

Suppose that 10,000 tickets are sold in one lottery and 5000 tickets are sold in another lottery. If a person owns 100 tickets in each lottery, what is the probability that she will win at least one first prize?

Two students A and B are both registered for a certain course. Assume that student A attends class 80 percent of the time, student B attends class 60 percent of the time, and the absences of the two students are independent. a. What is the probability that at least one of the two students will be in class on a given day? b. If at least one of the two students is in class on a given day, what is the probability that A is in class that day?

If three balanced dice are rolled, what is the probability that all three numbers will be the same?

Consider an experiment in which a fair coin is tossed until a head is obtained for the first time. If this experiment is performed three times, what is the probability that exactly the same number of tosses will be required for each of the three performances?

The probability that any child in a certain family will have blue eyes is 1/4, and this feature is inherited independently by different children in the family. If there are five children in the family and it is known that at least one of these children has blue eyes, what is the probability that at least three of the children have blue eyes?

Consider the family with five children described in Exercise 10. a. If it is known that the youngest child in the family has blue eyes, what is the probability that at least three of the children have blue eyes? b. Explain why the answer in part (a) is different from the answer in Exercise 10.

Suppose that A, B, and C are three independent events such that Pr(A) = 1/4, Pr(B) = 1/3, and Pr(C) = 1/2.(a) Determine the probability that none of these three events will occur. (b) Determine the probability that exactly one of these three events will occur.

Suppose that the probability that any particle emitted by a radioactive material will penetrate a certain shield is 0.01. If 10 particles are emitted, what is the probability that exactly one of the particles will penetrate the shield?

Consider again the conditions of Exercise 13. If 10 particles are emitted, what is the probability that at least one of the particles will penetrate the shield?

Consider again the conditions of Exercise 13. How many particles must be emitted in order for the probability to be at least 0.8 that at least one particle will penetrate the shield?

In the World Series of baseball, two teams A and B play a sequence of games against each other, and the first team that wins a total of four games becomes the winner of the World Series. If the probability that team A will win any particular game against team B is 1/3, what is the probability that team A will win the World Series?

Two boys A and B throw a ball at a target. Suppose that the probability that boy A will hit the target on any throw is 1/3 and the probability that boy B will hit the target on any throw is 1/4. Suppose also that boy A throws first and the two boys take turns throwing. Determine the probability that the target will be hit for the first time on the third throw of boy A

For the conditions of Exercise 17, determine the probability that boy A will hit the target before boy B does.

A box contains 20 red balls, 30 white balls, and 50 blue balls. Suppose that 10 balls are selected at random one at a time, with replacement; that is, each selected ball is replaced in the box before the next selection is made. Determine the probability that at least one color will be missing from the 10 selected balls.

Suppose that A1,...,Ak form a sequence of k independent events. Let B1,...,Bk be another sequence of k events such that for each value of j (j = 1, . . . , k), either Bj = Aj or Bj = Ac j . Prove that B1,...,Bk are also independent events. Hint: Use an induction argument based on the number of events Bj for which Bj = Ac j .

Prove theorem 2.2.2 on page 71. Hint: The “only if” direction is direct from the definition of independence on page 68. For the “if” direction, use induction on the value of j in the definition of independence. Let m = j – 1 and let l = 1 with j = i.

Prove Theorem 2.2.4 on page 73.

A programmer is about to attempt to compile a series of 11 similar programs. Let Ai be the event that the ith program compiles successfully fori = 1,..., 11. When the programming task is easy, the programmer expects that 80 percent of programs should compile. When the programming task is difficult, she expects that only 40 percent of the programs will compile. Let B be the event that the programming task was easy. The programmer believes that the events A1,...,A11 are conditionally independent given B and given Bc. a. Compute the probability that exactly 8 out of 11 programs will compile given B. b. Compute the probability that exactly 8 out of 11 programs will compile given Bc.

Prove Theorem 2.2.3 on page 72.