CLEP College Mathematics: Counting and Probability Practice Questions
Choose the right counting or probability model in eight original problems. Decide whether order matters, whether selections affect later selections and what information restricts the possible outcomes.
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Answer the questions below — you get instant feedback and a full explanation for each.
1. There are 4 shuttle routes from town A to town B and 3 routes from town B to town C. Every first route can connect with every second route. How many route pairs are possible for a trip from A through B to C?
Explanation. For each of the 4 first routes there are 3 second routes. The multiplication rule gives 4 times 3 = 12 route pairs. Adding would count individual route choices rather than complete two-part trips.
2. Seven volunteers will fill two different roles: chair and secretary. A person cannot hold both roles. How many assignments are possible?
Explanation. There are 7 choices for chair and then 6 remaining choices for secretary. The roles are different, so order matters: 7 times 6 = 42 assignments.
3. A three-person committee is chosen from 8 volunteers. There are no distinct roles on the committee. How many committees are possible?
Explanation. An ordered selection would give 8 times 7 times 6 = 336. Each committee appears in 3! = 6 orders, so divide by 6: 336/6 = 56.
4. A token is chosen uniformly from tokens numbered 1 through 12. What is the probability that its number is divisible by 3 or by 4, including numbers divisible by both?
Explanation. The qualifying numbers are 3, 4, 6, 8, 9 and 12, giving 6 of 12 equally likely outcomes. The probability is 6/12 = 1/2. Counting multiples of each separately would count 12 twice.
5. Six inspection folders are labeled A through F. Two different folders are selected uniformly at random, without replacement. What is the probability that the selected pair is exactly A and B, in either order?
Explanation. There are 6 choose 2 = 15 equally likely unordered pairs, and only {A, B} qualifies. Thus the probability is 1/15. Equivalently, the two possible orders give 2 times (1/6)(1/5).
6. Of 24 morning trainees, 9 cycle to class. Of 16 evening trainees, 4 cycle to class. A trainee is selected uniformly from all trainees who cycle. What is the probability the selected trainee attends in the morning?
Explanation. The condition 'who cycle' restricts the sample space to 9 + 4 = 13 trainees. Of those, 9 attend in the morning, so the conditional probability is 9/13. The denominator is not all 40 trainees.
7. Events A and B are independent, with P(A) = 4/5 and P(B) = 3/4. What is P(A and B)?
Explanation. For independent events, multiply the probabilities: (4/5)(3/4) = 12/20 = 3/5. Independence does not mean the events cannot happen together.
8. A spinner has four equally likely sectors worth 0, 2, 2 and 8 points. What is the expected number of points from one spin?
Explanation. Expected value is the probability-weighted average: (0 + 2 + 2 + 8)/4 = 3 points. It is a long-run average, so it need not be a possible score on a single spin.
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FAQ
When should I use combinations?
Use combinations when only the chosen members matter. Use ordered counting when assigning different roles or when a sequence changes the outcome.
What changes in a conditional probability?
The given condition restricts the sample space. Count qualifying outcomes within that restricted group.
Are independent events mutually exclusive?
These are different properties. Mutually exclusive events cannot both occur. Independent events do not change one another's probabilities.