← Apply probability distributions
Binomial probability P(X=r)=\binom{n}{r}p^{r}(1-p)^{n-r} n trials, r successes, probability p each. Poisson probability P(X=r)=\frac{e^{-\lambda}\lambda^{r}}{r!} λ = mean number of events in the interval.
Also check your table: the probabilities must sum to 1. Missing-value questions are usually solved by exactly that. Worked example Binomial A fair coin is tossed 5 times. P(exactly 3 heads)? n = 5, p = 0.5, r = 3. P = ⁵C₃ (0.5)³(0.5)² . ⁵C₃ = 10, and (0.5)⁵ = 1/32. P = 10 × 1/32 = 10/32 = 0.3125. ✓ P(3 heads) = 0.3125.
3 Discrete distributions
Watch: Discrete distributions Opens a YouTube search: pick a video that clicks for you| Distribution | Use when… | Mean | Variance |
|---|---|---|---|
| Uniform | all outcomes equally likely | (a+b)/2 | – |
| Binomial | fixed n trials, two outcomes, constant p, independent | np | np(1−p) |
| Poisson | events at a constant average rate λ in a fixed interval | λ | λ |
Spotting the distribution from the wording
- Poisson — a rate: look for “per” or “every” (calls per hour, faults per metre). Parameter λ. The rate must be proportional to the measure, so if the interval doubles, λ doubles.
- Binomial — a fixed number of trials with two outcomes: look for “out of”. Parameters n (trials) and p (probability of success).
- Normal — you are almost always told to use it. Parameters μ and σ.
Discrete random variables
Expected value E(X)=\sum x\,P(X=x) Multiply each value by its probability and add. This is the long-run mean, μ. Variance \mathrm{VAR}(X)=\sum x^2P(X=x)-[E(X)]^2 The “mean of the squares minus the square of the mean”. Standard deviation \mathrm{SD}(X)=\sqrt{\mathrm{VAR}(X)} Back into the original units. ✕E(X) does not have to be a possible value
The expected number of heads in 3 tosses is 1.5, and you cannot get half a head. E(X) is a long-run average, not a prediction. Do not round it to a whole number unless the question explicitly asks for a realistic count.Also check your table: the probabilities must sum to 1. Missing-value questions are usually solved by exactly that. Worked example Binomial A fair coin is tossed 5 times. P(exactly 3 heads)? n = 5, p = 0.5, r = 3. P = ⁵C₃ (0.5)³(0.5)² . ⁵C₃ = 10, and (0.5)⁵ = 1/32. P = 10 × 1/32 = 10/32 = 0.3125. ✓ P(3 heads) = 0.3125.