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.
Understanding Math: Probability with Combinations
A combination count becomes useful for probability only when every possible selection is equally likely. A well shuffled card deck is a good example because each possible hand of a fixed size has the same chance of appearing. Suppose a five card hand must contain exactly two hearts.
First count the ways to select two of the thirteen hearts. Then count the ways to select three of the thirty nine non-hearts. Multiplying these counts works because each valid hand needs one choice from each group.
The total number of hands comes from selecting any five cards from all fifty two cards. This method is often called splitting a condition into separate choices.
Words in a probability problem matter a great deal. Exactly two means that hands with one heart or three hearts do not qualify. At least two means that two, three, four, or five hearts qualify.
You can count each of those cases separately, but a complement can be faster. For example, to find the chance of getting at least one ace in a five card hand, it is easier to count hands with no aces. Those hands use only the forty eight non-ace cards.
Subtracting that probability from one gives the result. This approach avoids a long list of cases and reduces the chance of missing one.
Be careful when conditions overlap. If a committee must include a student from Year 10 or a student who plays football, some students may belong to both groups. Counting both groups separately can count the same committee twice.
A safe method is to define cases that cannot overlap, then add their counts. Another common mixed situation includes a choice followed by an assignment. Selecting three students for a committee uses combinations.
Choosing which selected student becomes chairperson adds a role, so order now matters at that stage. Separating the selection step from the role step makes the counting much clearer.
Large combination values can look intimidating, but you rarely need to calculate enormous factorials in full. Write the factorial products as consecutive factors, then cancel common factors before multiplying. Keep your work organised by stating what each number represents, such as total selections, selections meeting the condition, or selections excluded by a complement.
A final check is useful. The number of favourable selections cannot be larger than the total number of selections, so the probability must stay from zero to one.
The simple counting method only works when selections are equally likely. If a raffle gives some people extra tickets, or a machine produces outcomes with different chances, the outcomes need weighted probabilities instead of an ordinary combination ratio.
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.