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Chapter 2 Discrete Random Variables and Probability Distributions
4. X and Y play the following game: X tosses 2 fair coins and Y tosses 3. The player
throwing the greater number of heads wins. In case of a tie, the throws are repeated until a winner is determined.
a. What is the probability that X wins on the first play? b. What is the probability that X wins the game?
5. In a political race it is known that 40 of the voters favor candidate C. In a random
sample of 100 voters, what is the probability that
a. between 30 and 45 voters favor C? b. exactly 36 voters favor C?
6. A gambling game is played as follows. A player, who pays 4 to play the game,
tosses a fair coin 5 times. The player wins as many dollars as heads are tossed.
a. Find the probability distribution for N , the player’s net winnings. b. Find the mean and variance of the player’s net winnings.
7. A red die is fair, and a green die is loaded so that the probability it comes up 6
is
1 10
.
a. What is the probability of rolling exactly 3 sixes in 3 rolls with the red die? b. What is the probability of at least 30 sixes in 100 rolls of the red die?
c. The green die is thrown 5 times and the red die is thrown 4 times. Find the
probability that a total of 3 sixes occurs.
8. What is the probability of one head twice in 3 tosses of 4 fair coins? 9. A commuter’s drive to work includes 7 stoplights. Assume the probability that a
light is red when the commuter reaches it is 0.20, and that the lights are far enough apart to operate independently.
a. If X is the number of red lights the commuter stops for, find the probability
distribution function for X.
b. Find P. X ≥ 5: c. Find P. X ≥ 5 | X ≥ 3:
10. The probability of being able to log on a computer system from a remote terminal
during a busy period is 0.7. Suppose that 10 independent attempts are made and that X denotes the number of successful attempts.
a. Write an expression for the probability distribution function, f .x. b. Find P. X ≥ 5:
c. Now suppose that Y represents the number of attempts up to and including the
first successful attempt. Write an expression for the probability distribution function, g.y.
2.5 A Recursion
