Probability, Combinatorics and Simple Counting
Probability items in EPSO tests stay at the level of counting outcomes and combining simple events. The difficulty is choosing between "and" and "or", and knowing whether order matters.
Core rules
- P(event) = favourable outcomes / total equally likely outcomes.
- Complement: P(at least one) = 1 − P(none). This is almost always the faster route.
- Independent events ("and"): multiply. Two independent checks each catching 90% of defects means both catch 0.9 × 0.9 = 0.81.
- Mutually exclusive events ("or"): add.
- Without replacement: the denominator, and sometimes the numerator, falls by one on each draw.
- Combinations (order irrelevant): C(n,r) = n!/(r!(n−r)!). Permutations (order matters): P(n,r) = n!/(n−r)!.
- Arrangements with repeats: n! divided by the factorial of each repeat count.
Worked example
A panel of 3 is drawn from 5 women and 3 men. What is the probability that it contains exactly one man?
Total panels: C(8,3) = 56. Favourable: choose 1 man from 3 and 2 women from 5, so C(3,1) × C(5,2) = 3 × 10 = 30. P = 30/56 = 15/28 ≈ 0.536.
The "and" inside the favourable count is a product; the total is a combination because the order of selection does not create a new panel.
Sanity checks
Every probability lies between 0 and 1 - an answer above 1 means you added where you should have multiplied. The probability of "at least one" must rise as the number of trials rises; if it falls, the complement has been mis-set.
Traps
- Adding probabilities of independent events that must all occur.
- Keeping the original denominator when drawing without replacement.
- Using permutations when the question describes a committee, a sample or a set.
- Forgetting that "at least one" includes the cases of two, three and more, which is precisely why the complement is used.