Probability Practice Problems and Solutions

Problem 1 - Types of Exams and First Question Difficulty Probability

1. A professor gives only two types of exams, “easy” and “challenging”. The probability of a challenging exam is 0.80. The probability that the first question on the exam will be difficult is 0.90 if the exam is challenging and is 0.15 if it is easy.

a. What is the probability that the first question on your exam is difficult?
b. What is the probability that your exam is challenging, given that the first question on the exam is difficult?

2. A bridge hand in which there is no card higher than a nine is called a Yarborough. Specify an appropriate sample space and determine the probability of Yarborough when you are randomly dealt 13 cards out of a well-shuffled deck of 52 cards.

3. The probability that a visit to a particular car dealer results in neither buying a second-hand car nor a Japanese car is 55%. Of those coming to the dealer, 25% buy a second-hand car and 30% buy a Japanese car. What is the probability that a visit leads to buying a second-hand Japanese car?

4. A gambling book recommends the following winning strategy for a game of roulette: It recommends that the gambler bet $1 on red. If red appears (which happened with probability !"#" ) then the gambler should take her $1 and quit. If the gambler loses this bet (which might occur with probability $% #"), she should make an additional $1 bets on red on each of the two next spins of the roulette wheel, and then quit. Let X denote the gambler’s winnings when she quits.

a. Find 0).
b. Are you convinced that the recommended strategy is indeed a winning strategy?

5. You and two of your friends are in a group of 10 people. The group is randomly split up into two groups of 5 people each. Specify an appropriate sample space and determine the probability that you and your two friends are in the same group.

6. A bridge hand in which there is no card higher than a nine is called a Yarborough. Specify an appropriate sample space and determine the probability of Yarborough when you are randomly dealt 13 cards out of a well-shuffled deck of 52 cards.

7. There are eight balls in a box. Each ball has been colored red or white with equal probabilities, independently of other balls. You see that two red balls are added to the box. Afterwards, five balls ball are taken at random and are shown and all of these five balls are white. What is the probability that all the other five balls in the bowl are red?

8. If A, B and D are three events such that P(A ? B ? D) = 0.7, what is the value of P(A& ? B& ? D&)?

9. A and B are disjoint. What is the condition for A& and B& to be disjoint as well?

10. A group of 3 girls and 4 boys go to the local movie theater together, and sit in a row of 10 seats. Assume that each seating arrangement is equally likely.

a. If no other person is sitting in the row except for the group, how many possible seating arrangements are there in total?
b. The group was asked to move to a row with only 7 seats. Find the probability that no two people of the same gender sit next to each other.
c. The group is in a row with 7 seats. Find the probability that all the boys sit next to each other.

11. Two balls are chosen randomly from an urn containing 8 white, 4 black, and 2 red balls. Suppose that we win $2 for each black selected and we lose $1 for each white ball selected. Let X denote our winnings.

a. Determine the probability mass function for X.
b. Determine the cumulative distribution function for X defined in this problem and sketch its graph.
c. Find E[X].