Probability and Statistics: Chance, Conditional Probability and Inference
Core concepts
You need solid command of sample spaces, events and the classical, relative-frequency and subjective interpretations of probability. Know the addition rule for mutually exclusive events, the multiplication rule for independent events, and the complement rule P(A′) = 1 − P(A). Be confident with two-way tables and tree diagrams, including conditional probability P(A|B) = P(A ∩ B)/P(B), and with the language of independence: A and B are independent when P(A ∩ B) = P(A)P(B), equivalently P(A|B) = P(A). Understand discrete random variables through their probability distributions, expected value E(X) = Σ x·P(x) and variance. For statistical inference, interpret a confidence interval as a range of plausible values for the population parameter, and understand that a larger sample narrows the interval.
Traps to watch
- Adding probabilities of events that are not mutually exclusive without subtracting the overlap.
- Assuming independence when events are only described as separate.
- Reading a conditional probability in the wrong direction - P(A|B) is not P(B|A).
- Treating a confidence interval as a probability that the parameter lies inside it.
- Forgetting to convert a percentage relative frequency into a proportion before multiplying.
Worked example
In a class, 60% study Biology, 40% study Chemistry, and 25% study both. Then P(B ∪ C) = 0.60 + 0.40 − 0.25 = 0.75. The conditional probability P(B|C) = 0.25 ÷ 0.40 = 0.625, so a Chemistry student has a 62.5% chance of also studying Biology. Since 0.25 ≠ 0.60 × 0.40 = 0.24, the events are not independent.
Rule of thumb
Draw the two-way table or tree first; write the condition after the vertical bar as the denominator of the fraction.