Probability, at a glance.
A compact refresher on the probability ideas worth keeping close.
An original Bimankik study aid for general learning. This is not official examination material and does not replace your examination body’s current syllabus or approved reference tables.
Start with the event.
A probability is a number between 0 and 1 describing how likely an event is under a specified model. Before choosing a formula, write down what your event actually means and the assumptions you are making.
The essentials
| Idea | Relationship | Watch out for |
|---|---|---|
| Complement | P(Aᶜ) = 1 − P(A) | Does “not A” include every other possible outcome? |
| Union | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) | Subtract the overlap to avoid counting it twice. |
| Conditional probability | P(A | B) = P(A ∩ B) / P(B) | Requires P(B) > 0. |
| Independence | P(A ∩ B) = P(A)P(B) | Not the same as mutually exclusive events. |
| Bayes’ rule | P(A | B) = P(B | A)P(A) / P(B) | Keep the conditioning direction clear. |
Expected value and variance
For a discrete random variable X, E[X] = Σ x P(X = x). For a continuous variable with density f, replace the sum with the appropriate integral of x f(x).
Var(X) = E[X²] − (E[X])². For constants a and b, E[aX + b] = aE[X] + b and Var(aX + b) = a²Var(X).
Linearity of expectation does not require independence: E[X + Y] = E[X] + E[Y]. For variance, the covariance matters: Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y).
A quick worked example
Suppose 20% of a portfolio belongs to group A. A claim is reported by 10% of group A and 5% of the remaining group. The probability that a randomly chosen policy reports a claim is 0.2 × 0.1 + 0.8 × 0.05 = 0.06.
Given that a claim was reported, the probability it came from group A is (0.2 × 0.1) / 0.06 = 1/3. Notice that this is not the original 20% portfolio share: the evidence changed the conditional probability.
Before you move on
- Define the event and the conditioning information.
- Check whether independence has been stated or justified.
- Keep probabilities within [0, 1].
- Sanity-check whether the result fits the situation.