5. The following is a sample space for the experiment of tossing a coin three times and noting whether heads or tails comes up.

HHH- HHT- HTH- HTT -THH- THT -TTH -TTT

Let Abe the event that the first two tosses are different (one heads and one tails, but not necessarily in that order) and let B be the event that exactly two of the three tosses come up heads. Calculate the following probabilities. (Assume that heads and tails have equal probability of coming up on any given toss.)

a) What is P(A)?

b) What is P(B)?

c) What is P (AandB)?

d) What is P(AorB)?

e) What is P(B|A), the conditional probability of B given A?

Questions 6 and 7 refer to the following experiment. A vase contains ten marblesthree are red and seven are black. Two marbles are drawn from the vase in succession, and the color of each marble is recorded. (The first marble is not returned to the vase before the second is drawn. You may assume that all marbles in the vase at a given time have equal probability of being drawn.)

6. Calculate the following probabilities.

a) What is the probability that both marbles chosen are black?

b) What is the probability that both marbles chosen are red?

c) What is the probability that at least one marble chosen is red?

d) What is the probability that exactly one marble chosen is red?

7. Referring to the same experiment as in Question 6, let X be the number of red marbles selected when this experiment is run.

a) Find the probability distribution of the random variable X. Write your answer in the form of a table. (Hint: The calculations in Question 6 should help you here.)

b) Calculate the meanof the random variable X. (You can use the table in part (a) in your calculations. Show your work in any case.)

c) Calculate the standard deviationof the random variable X. (Write your answer to to two decimal places. Show your work.)