Exercises - Counting and Probability

  1. A 4-member micro team must be selected from a group of 10 soccer players. How many different micro teams are possible?

  2. How many different 5-digit odd numbers are possible if digits may be repeated?

  3. A question on a history quiz requires that 6 events be arranged in the proper chronological order. How many arrangements are possible?

  4. How many different groupings of 6 females are possible from an applicant pool of 20?

  5. What is the probability of getting at least one head in the tossing of a coin 3 times?

  6. Assume two fair dice are rolled.

    1. What is the probability of rolling a sum of 7 or 11?
    2. What is the probability that the sum is a prime number?
    3. What is the probability that the sum is greater than 7 if you already know that the number showing on one die is 3?
    4. What is the probability that the sum is at least 7?
    5. What is the probability that the sum is even and is greater than 7?
  7. How many 4-digit even numbers are there if the digits may not be repeated?

  8. Four people are being considered for promotion from a pool of qualified applicants made up of 20 females and 15 males. The four selected were all males. What is the probability of this situation happening by chance?

  9. How many different orderings of letters can be made from the following?

    AAABBBBCCCCCDDDDD
  10. How many different subcommittees of 4 can be formed from a club of 24 members?

  11. A true/false test has 10 items. How many different answer keys are possible?

  12. How many different 5-digit odd numbers are possible if a digit may not be repeated?

  13. How many different arrangements of letters are possible in the word BOOKKEEPER?

  14. If it is known that 80% of all freshmen have had at least one drink, what is the probability of the following if 5 freshmen are selected at random?

    1. all 5 freshmen have had at least one drink
    2. at least one freshman has not had at least one drink
  15. A jury of 12 is to be selected from a list of 35 males and 15 females. What is the probability that there is no more than one female on the jury?

  16. You are in a room of 12 people. What is the probability that at least two of these people will have the same birthday?

  17. If the digits may be repeated,

    1. How many four-digit odd numbers are less than 5000?
    2. How many three-digit even numbers, larger than 399, are there?
  18. A multiple choice test has 20 items. Each item has 3 choices. How many distinctly different answer keys are possible?

  19. A box contains 8 good light bulbs and 4 that are defective. If three bulbs are randomly selected from the box without replacement, find the probability that at least one will be defective.

  20. Five people are to be randomly selected for a committee from a class consisting of 10 freshmen and 8 sophomores. What is the probability that all five of those selected are sophomores?

  21. On a multiple choice test with answers a, b, c, d -- what is the probability of getting the first three questions correct (assuming you know nothing about the questions)?

  22. There are 3 defective calculators in a box of 100. Out of five selected, what is the probability that:

    1. four are good and one is defective
    2. there are at most two defective
    3. all five are good
  23. Six red balls and four blue balls are placed in an urn. Four balls are drawn at random from the urn. Find the probability that all four are blue if:

    1. each ball is returned to the urn before the next is drawn
    2. the balls are not returned to the urn
  24. A committee of six is randomly selected from a club with 18 male and 12 female members

    1. What is the probability that there is at least one female on the committee?
    2. Find the probability that there are three males and three females on the committee.
  25. Toss a pair of tetrahedron dice, each numbered 1,2,3,4. Let $X$ be the sum of the numbers. Find the probability of getting a sum that is:

    1. at least 6
    2. at most 4
    3. more than 6
    4. less than 4
    5. an even number
    6. a prime number
    7. divisible by 4
  26. Company A supplies 40% of the computers sold and is late 5% of the time. Company B supplies 30% of the computers sold and is late 3% of the time. Company C supplies another 30% and is late 2.5% of the time. A computer arrives late; what is the probability that it came from Company A?