Counting and combined probability
Decide whether order matters and whether events are independent.
What you will learn
- Use combinations for unordered selections.
- Distinguish independent from mutually exclusive events.
Count outcomes consistently
A committee is a selection, so rearranging its members does not create a new committee. Use combinations to count selections. In contrast, choosing a first, second and third place distinguishes order and needs a different count.
A probability can be calculated as favourable outcomes divided by all outcomes only when the counted outcomes are equally likely and use a consistent counting method.
Independence is a condition
Independent events satisfy P(A∩B)=P(A)P(B). Mutually exclusive events cannot happen together, so their intersection has probability zero. Events with positive probabilities cannot be both independent and mutually exclusive.
For any two events, P(A∪B)=P(A)+P(B)−P(A∩B). Subtract the intersection so outcomes in both events are not counted twice.
Worked example
A team of 2 is selected from 7 students. How many teams are possible?
Show the worked solution
- Order does not matter: choosing Ana then Ben is the same team as Ben then Ana.
- There are 7×6 ordered selections.
- Divide by 2! to remove duplicate orderings.
Answer 21 teams
Common mistakes
- Do not multiply event probabilities unless independence or the appropriate conditional probability justifies it.
- A committee is not an ordered arrangement.
What can you explain now?
Independent events have probabilities 0.3 and 0.5. Find the probability that both occur.
Compare with the explanation
0.15
Independence permits multiplication: 0.3×0.5=0.15.
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These lessons teach the topics represented in our current practice sets. They are not a complete course for every paper or option in the qualification.
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