Section PC.8 – Probability with Permutations and Combinations Practice Problems

1. What is the probability that if 6 letters are typed, no letters are repeated?

2. 4 letters are typed, without repetition. What is the probability that all 4 will be vowels? Write your answer as a percent. Round your answer to three decimal places.

3. Four letters are typed, without repetition. What is the probability that all 4 will be vowels?

4. Five letters are typed, with repetition allowed. What is the probability that all 5 will be vowels?

5. Ten horses are in a race.  You are trying to predict who will finish 1st , 2nd, and 3rd place.  What is the probability you choose all three places correctly? Express the answer as a simplified fraction.

6. You own 16 CDs that are stored in a box. You have decided to display 5 of them on a shelf.  One at a time, you randomly select a CD from the box and place it on the shelf until there are 5 CDs on the shelf.  What is the probability that the CDs on the shelf end up in alphabetical order?

7. 4 letters are typed, with repetition allowed. What is the probability that all 4 will be vowels? Write your answer as a percent. Round to the nearest hundredth of a percent as needed.

8. A jury pool consists of 27 people, 14 men and 13 women. Compute the probability that a randomly selected jury of 12 people is all male.

9. A jury pool consists of 27 people, 14 men and 13 women. Compute the probability that a randomly selected jury of 12 people has 6 males and 6 females. Express answer as a simplified fraction.

10. 5 similar cars are entered in a race. The cars all have an equal chance of winning and there are no ties. Spectators are invited to complete a single prize ticket with their guesses for which cars will finish in first, second, and third place. The spectators who correctly guess the first, second, and third place finishers will get a small prize.

a. How many different ways can a prize ticket be completed?
b. What is the probability that a spectator will wins the small prize? Write the answer as a simplified fraction.
c. Suppose spectators are allowed to complete 2 different prize tickets. What is the probability that a spectator, with 2 prize tickets, will wins the small prize? Write the answer as a simplified fraction.

11. In a lottery game, a player picks six numbers from 1 to 48. If 5 of the 6 numbers match the winning numbers, then the player wins the second prize. What is the probability of winning this prize?

12. In a lottery game, a player picks six numbers from 1 to 48. If 4 of the 6 numbers match those drawn, they player wins third prize. What is the probability of winning this prize?

13. A housing development offers homes with four different options. The homes were built with one choice from each of the options below:

• Number of Bedrooms: Two Bedrooms, Three Bedrooms, or Four Bedrooms
• Number of Bathrooms: One Bathroom, Two Bathrooms, or Three Bathrooms
• Number of Floors: One Floor or Two Floors
• Type of Yard: Grass or Desert Landscaping
• There are an equal number of houses with each combination of options

You would like to buy a house with three bedrooms or four bedrooms, one bathroom, one floor or two floors, and desert landscaping. If there is only one house left to buy, what is the probability that it has what you are looking for?

14. Compute the probability that a 5-card poker hand is dealt to you that contains all hearts.

15. Compute the probability that a 5-card poker hand is dealt to you that contains four Aces.

16. Compute the probability that a 5-card poker hand is dealt to you that contains three kings and two sevens. Express answer as simplified fraction.

17. Eliana, Gil, Erika, Lucille, Gabriel, Emma, and Cristian have all been invited to a dinner party. They arrive randomly and each person arrives at a different time. Find the probability that Emma arrives first and Cristian arrives last. Write the answer as a simplified fraction.

18. Eliana, Gil, Erika, Lucille, Gabriel, Emma, and Cristian have all been invited to a dinner party. They arrive randomly and each person arrives at a different time. Find the probability that Emma arrives first and Cristian arrives last. Write the answer as a simplified fraction.

a. Determine the probability that exactly 3 of these cards are Aces.
b. Determine the probability that all five of these cards are Spades.
c. Determine the probability that exactly 3 of these cards are face cards.
d. Determine the probability of selecting exactly 2 Aces and exactly 2 Kings.
e. Determine the probability of selecting exactly 1 Jack.

19. What is the probability that if 4 letters are typed, no letters are repeated? Write your answer in decimal form, rounded to the nearest thousandth.

20. A chef is choosing from 12 different entrees for an important banquet, 3 of which contain spinach. The guests will have a choice of 4 entrees.  If the chef chooses those 4 entrees at random, find the probability, rounded to the nearest thousandth, that:

a. None contain spinach.
b. Exactly one contains spinach.
c. Exactly three have spinach.
d. All 4 have spinach.

21. A combination lock has 40 numbers on it, from zero to 39.  Find the probability that if the combination to unlock it consists of three numbers, it must contain the numbers 1, 2, and 3 in some order.

a. Assume that numbers cannot be repeated in the combination. Express the answer as a simplified fraction.
b. Assume that numbers can be repeated in the combination. Express the answer as a simplified fraction.

22. A box contains 25 calculators, 6 of which are defective.  If 6 are selected at random, find the probability that:

a. All are defective.
b. None are defective.

Answer both as a simplified fraction and as a decimal with 9 decimal places.

23. What is the probability (without replacement) you will draw the ace of spades first, the king of spades second, and the queen of spades third? Express the answer as a simplified fraction.

24. What is the probability (without replacement) you will draw the ace of spades first, the king of spades second, and the queen of spades third? Express the answer as a simplified fraction.