Probabilistic Reasoning
What this domain tests
Probabilistic Reasoning examines random experiments, sample spaces, events, probability, odds, and expected value. The exam expects you to enumerate outcomes correctly, compute probabilities, and interpret odds and expected value in decision contexts.
Core rules and concepts
- Random experiment and sample space: list all possible outcomes. For two independent stages, multiply the number of outcomes at each stage.
- Events: the probability of an event $A$ is $P(A)=\frac{\text{number of favourable outcomes}}{\text{total number of equally likely outcomes}}$. The complement satisfies $P(A')=1-P(A)$.
- Combined events: for mutually exclusive events, add probabilities; for independent events, multiply. For events that are not mutually exclusive, subtract the overlap: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$.
- Odds: odds in favour of $A$ are $\frac{P(A)}{P(A')}$; odds against are the reciprocal. Convert odds to probability with $P(A)=\frac{\text{favourable}}{\text{favourable}+\text{unfavourable}}$.
- Expected value: $E=\sum (\text{value}\times\text{probability})$. A fair game has expected value zero for the player.
Worked example
A game costs $2 to play. A fair six-sided die is rolled: rolling a 6 pays $10, rolling a 5 pays $4, and any other result pays nothing. Is the game fair?
Sample space: six equally likely outcomes. The payouts and probabilities are: $10 with probability $\frac{1}{6}$, $4 with probability $\frac{1}{6}$, and $0 with probability $\frac{4}{6}$.
Expected payout: $10\times\frac{1}{6}+4\times\frac{1}{6}+0\times\frac{4}{6}=\frac{10}{6}+\frac{4}{6}=\frac{14}{6}\approx2.33$ dollars.
Net expected value for the player: $2.33-2=0.33$ dollars. Since this is positive, the game slightly favours the player, so it is not fair to the operator.
Traps that cost marks
- Treating outcomes as equally likely when they are not (for example, sums of two dice).
- Adding probabilities of independent events instead of multiplying them.
- Forgetting to subtract the overlap for non-mutually exclusive events.
- Confusing odds with probability: odds of 3:1 correspond to a probability of $\frac{3}{4}$, not 3.
- Omitting the cost of playing when computing net expected value in a game.