Probability with combinations is used when you need to count outcomes where order does not matter. It helps answer questions about card hands, lottery drawings, committees, and selections from a group. The main idea is to compare the number of favorable outcomes to the total number of possible outcomes.
This gives a probability between 0 and 1, or between 0% and 100%.
Combinations are useful because listing every outcome can be too slow or impossible for large situations. The formula nCr = n! / (r!(n - r)!) counts how many ways to choose r objects from n objects without caring about order. Once the total outcomes and favorable outcomes are counted correctly, use P(event) = favorable outcomes / total outcomes.
Many mistakes happen when students use permutations instead of combinations or count the same selection more than once.
Key Facts
- Probability formula: P(event) = favorable outcomes / total outcomes
- Combination formula: nCr = n! / (r!(n - r)!)
- Permutation formula: nPr = n! / (n - r)!
- Use combinations when order does not matter, such as a 5-card hand.
- Use permutations when order matters, such as 1st, 2nd, and 3rd place.
- Complement rule: P(not A) = 1 - P(A)
Vocabulary
- Probability
- A number from 0 to 1 that describes how likely an event is to happen.
- Combination
- A selection of objects where the order of the chosen objects does not matter.
- Permutation
- An arrangement or selection of objects where the order does matter.
- Favorable outcome
- An outcome that satisfies the event or condition being studied.
- Sample space
- The set of all possible outcomes in a probability situation.
Common Mistakes to Avoid
- Using permutations when order does not matter. This is wrong because it counts the same group multiple times, such as treating ABC and CBA as different hands.
- Forgetting that the denominator must count all possible outcomes. This is wrong because probability must compare favorable outcomes to the entire sample space.
- Mixing independent and without-replacement situations. This is wrong because probabilities change after an item is removed and not replaced.
- Counting favorable outcomes too narrowly or too broadly. This is wrong because the numerator must match exactly the event described in the problem.
Practice Questions
- 1 A bag has 8 red marbles and 6 blue marbles. If 3 marbles are chosen at random without replacement, what is the probability that all 3 are red?
- 2 From a standard 52-card deck, what is the probability of being dealt exactly 2 hearts in a 5-card hand?
- 3 A lottery asks players to choose 6 numbers from 1 to 40, and the order drawn does not matter. Explain why combinations are used instead of permutations to count the possible tickets.