Formula sheet: Probability

Probability14 min readLesson 1 of 8

1.1 How to use this sheet This page carries the whole working equipment of the chapter. The definition of conditional probability is the source of everything else, so learn it first and learn which event goes into the denominator. The multiplication theorem and the test for independence follow directly. The theorem of total probability and Bayes' theorem come next, and they are the two results that turn case-by-case information into a single answer. The last three blocks cover distributions: the mean and variance of a random variable, and the binomial formula with its own mean and variance. The formula blocks below are labelled by the job they do. Read the labelled block first, then the short description, and finally practise the formula on one worked example from the questions page. A formula that has been used once is remembered far better than a formula that has only been read. 1.2 Which answer each block supports When a question gives a condition and asks for a probability, the first block is the tool. When it gives stepwise information and asks for a joint probability, the second block is the tool. When the word independent appears, the fourth block decides the calculation. When a question describes cases that produce an observed result, the fifth and sixth blocks are the two directions of the same reasoning. For random variables, the seventh and eighth blocks give the mean and spread, and the last two blocks handle repeated trials with two outcomes. 1.3 A five-line summary to write on a blank page Condition first, then the joint, then the test for independence, then the two case formulas, then the distribution formulas. If all five lines can be reproduced without looking, the sheet has done its job, and the questions page will only be arithmetic.

P (E ∣ F) = \frac{P ( E \cap F )}{P ( F )}, P (E \cap F) = P (E) P (F ∣ E)

The definition and its rearrangement are the first pair to memorise.

P (E ∣ F) = \frac{P ( E \cap F )}{P ( F )}, P (F) \neq = 0

Conditional probability: the probability of E once F is known to have happened.

P (E \cap F) = P (E) P (F ∣ E) = P (F) P (E ∣ F)

Multiplication theorem for two events.

P (E \cap F \cap G) = P (E) P (F ∣ E) P (G ∣ E \cap F)

Chain rule for three events; each factor conditions on everything before it.

P (E \cap F) = P (E) P (F)

Test for independence of two events.

P (E) = P (E_{1}) P (E ∣ E_{1}) + P (E_{2}) P (E ∣ E_{2}) + \dots + P (E_{n}) P (E ∣ E_{n})

Theorem of total probability, with E1 to En a partition of the sample space.

P (E_{i} ∣ E) = \frac{P ( E _{i} ) P ( E ∣ E _{i} )}{\sum _{j} P ( E _{j} ) P ( E ∣ E _{j} )}

Bayes' theorem: reversing the conditioning after E is observed.

μ = i \sum x_{i} p_{i}

Mean of a random variable with probability distribution pi.

σ^{2} = i \sum (x_{i} - μ)^{2} p_{i} = i \sum x_{i}^{2} p_{i} - μ^{2}

Variance, with the second form usually quicker.

P (X = r) =^{n} C_{r} p^{r} q^{n - r}, r = 0, 1, \dots, n

Binomial distribution for n Bernoulli trials, q = 1 - p.

μ = n p, σ^{2} = n pq

Mean and variance of the binomial distribution.

Exam

Write the event that comes after the bar first, because that is the event whose outcomes form the new sample space. In Bayes' theorem the denominator is shared by every numerator, so compute it once. Never assume independence; test it with the multiplication condition. In distribution questions, the sum of the probabilities is the first line of every solution.

in hinglish

Is chapter ka poora kaam ek hi definition se chalta hai: conditional probability mein bar ke baad wala event naya sample space banata hai, isliye usi ko denominator mein likhna hai. Uske baad multiplication theorem se joint probability milti hai, aur independent events ke liye product wala test hai. Jab cases diye hon aur unka mila-jula result poochha jaaye to total probability ka sum use karo; ulta sawaal, yaani result se case poochna ho to Bayes' theorem lagao. Random variable ke liye mean weighted average hota hai aur variance mean se doori ka weighted square; binomial distribution mein ek hi formula se exactly r successes ki probability nikal jaati hai, aur uske mean aur variance seedhe np aur npq hote hain. Formula sheet ke blocks ko labels ke saath padho, phir questions page par ek example par use karo; isse formula yaad rehta hai.