CA Foundation · Quantitative Aptitude

Probability: formula sheet

Full chapter guide

Key formulas

Classical probability: P(A) = m ÷ n; m = outcomes favourable to A, n = total outcomes in S. Valid only when outcomes are equally likely and exhaustive.
Statistical probability: P(A) = limit of (f ÷ n) as n → ∞; f = number of times A occurs in n trials. In practice you use a large n as an estimate.
Axiom 1 (non-negativity): 0 ≤ P(A) ≤ 1; Holds for every event A.
Axiom 2 (certainty): P(S) = 1; The sure event has probability 1.
Axiom 3 (additivity): P(A ∪ B) = P(A) + P(B); Only for mutually exclusive events A and B.
Complement rule: P(A′) = 1 − P(A); Useful for 'at least one' questions.
Odds: Odds in favour of A = m : (n − m); Then P(A) = m ÷ n. Odds against = (n − m) : m.
Exhaustive and exclusive set: P(A₁) + P(A₂) + ... + P(Aₖ) = 1; For events that are mutually exclusive and exhaustive.
Classical probability: P(E) = m ÷ n; m = favourable outcomes, n = total outcomes. Valid only when all outcomes are equally likely.
Range of probability: 0 ≤ P(E) ≤ 1; A result above 1 or below 0 means you have made an error.
Complement: P(not E) = 1 − P(E); Use it for 'at least one' questions: P(at least one) = 1 − P(none).
Odds in favour: m : (n − m); Favourable : unfavourable.
Odds against: (n − m) : m; Unfavourable : favourable.
Probability from odds: If odds in favour are a : b, P(E) = a ÷ (a + b); If odds against are a : b, then P(E) = b ÷ (a + b).
Combinations: nCr = n! ÷ [r! × (n − r)!]; Use when order of selection does not matter. Also nCr = nC(n − r).
Permutations: nPr = n! ÷ (n − r)!; Use when order matters.
Size of sample space for coins and dice: n coins: 2ⁿ outcomes. n dice: 6ⁿ outcomes.; Two dice: 36. Three coins: 8. Three dice: 216.
Pack of cards facts: 52 cards = 4 suits × 13 cards. 26 red, 26 black. 12 face cards (J, Q, K). 4 aces.; Each suit has 13 cards: A, 2 to 10, J, Q, K.
Drawing r items from a mixed group: P = (aCx × bCy) ÷ (a+b)C(x+y); Choose x from group a and y from group b, without replacement, out of a + b items in total.
Addition theorem (general): P(A ∪ B) = P(A) + P(B) − P(A ∩ B); Works for any two events A and B.
Mutually exclusive events: P(A ∪ B) = P(A) + P(B); Use only when A and B cannot occur together, so P(A ∩ B) = 0.
Complement rule: P(A') = 1 − P(A); A' means A does not occur.
Neither A nor B: P(A' ∩ B') = 1 − P(A ∪ B); Probability that none of the two events occurs.
Three events: P(A ∪ B ∪ C) = P(A) + P(B) + P(C) − P(A ∩ B) − P(B ∩ C) − P(A ∩ C) + P(A ∩ B ∩ C); If the three are pairwise mutually exclusive, just add the three probabilities.
Only A (not B): P(A ∩ B') = P(A) − P(A ∩ B); Useful for "A but not B" questions.
Exactly one of A, B: P(exactly one) = P(A) + P(B) − 2P(A ∩ B); Counts A only and B only, excluding both.
General multiplication theorem: P(A ∩ B) = P(A) × P(B | A) = P(B) × P(A | B); Works for any two events. Use it when events are dependent.
Multiplication rule for independent events: P(A ∩ B) = P(A) × P(B); Use only when A and B are independent.
Test for independence: A and B are independent ⇔ P(A ∩ B) = P(A) × P(B); Equivalent to P(B | A) = P(B) when P(A) is non-zero.
Three events (independent): P(A ∩ B ∩ C) = P(A) × P(B) × P(C); All events must be independent of each other.
At least one of independent events: P(at least one) = 1 − P(A′) × P(B′); Complements of independent events are also independent.
Mutually exclusive events: P(A ∩ B) = 0; If both P(A) and P(B) are above 0, such events are not independent.
Conditional probability: P(A | B) = P(A ∩ B) ÷ P(B); Valid only when P(B) > 0. B is the given (condition) event.
Reverse conditional: P(B | A) = P(A ∩ B) ÷ P(A); Valid only when P(A) > 0. Not the same as P(A | B) in general.
Multiplication rule: P(A ∩ B) = P(B) × P(A | B) = P(A) × P(B | A); Use it for draws without replacement. Multiply the first probability by the updated second one.
Equally likely outcomes: P(A | B) = n(A ∩ B) ÷ n(B); Count only inside B. Works when all outcomes are equally likely.
Complement given B: P(A' | B) = 1 − P(A | B); The condition B must stay the same on both sides.
Test for independence: A and B are independent if P(A | B) = P(A), that is, P(A ∩ B) = P(A) × P(B); If P(A | B) differs from P(A), the events are dependent.
Conditional probability: P(A | B) = P(A ∩ B) ÷ P(B); Valid when P(B) > 0.
Multiplication rule: P(A ∩ B) = P(A) × P(B | A); This gives each branch of a probability tree.
Total probability theorem: P(B) = P(A₁)P(B | A₁) + P(A₂)P(B | A₂) + ... + P(Aₙ)P(B | Aₙ); A₁ to Aₙ must be mutually exclusive and exhaustive.
Bayes' theorem: P(Aᵢ | B) = P(Aᵢ)P(B | Aᵢ) ÷ [P(A₁)P(B | A₁) + ... + P(Aₙ)P(B | Aₙ)]; The denominator is P(B) from the total probability theorem. It needs P(B) > 0.
Sum of posteriors: P(A₁ | B) + P(A₂ | B) + ... + P(Aₙ | B) = 1; Use this as a quick check on your answers.
Expected value (discrete): E(X) = Σ x·p(x) = x₁p₁ + x₂p₂ + … + xₙpₙ; Probabilities must be non-negative and satisfy Σ p = 1.
Expectation of a function: E(X²) = Σ x²·p(x); Square the value x, not the probability. Do not use [E(X)]² here.
Variance using expectation: Var(X) = E(X²) − [E(X)]²; Standard deviation is the positive square root of the variance.
Expectation of a constant: E(c) = c; A constant always has the same value, so its average is itself.
Linear change: E(aX + b) = a·E(X) + b; Holds for any constants a and b. The constant b is not multiplied by a.
Addition theorem: E(X + Y) = E(X) + E(Y); Holds for any two random variables, independent or not.
Multiplication theorem: E(XY) = E(X)·E(Y); Holds only if X and Y are independent.
Variance under linear change: Var(aX + b) = a²·Var(X); Adding b does not change the spread.
Fair game: Fair if expected gain = 0; Expected gain is the sum of (gain or loss × probability), with losses taken as negative.

Quick revision

Probability always lies between 0 and 1; P(impossible event) = 0 and P(sure event) = 1.
Classical: P(A) = favourable outcomes ÷ total outcomes, only when outcomes are equally likely.
P(not A) = 1 − P(A). Use it for "at least one" questions.
Mutually exclusive events cannot happen together: P(A ∪ B) = P(A) + P(B).
General addition rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Independent events: P(A ∩ B) = P(A) × P(B). Mutually exclusive events with non-zero probabilities are not independent.
General multiplication rule: P(A ∩ B) = P(A) × P(B | A).
Conditional probability: P(A | B) = P(A ∩ B) ÷ P(B), for P(B) > 0.
Bayes: P(Aᵢ | B) = P(Aᵢ) × P(B | Aᵢ) ÷ Σ[P(Aⱼ) × P(B | Aⱼ)], where the Aᵢ are mutually exclusive and exhaustive causes.
Expectation: E(X) = Σ x × P(x). It is a weighted average, not necessarily a value X can take.
A pack has 52 cards: 4 suits of 13, 12 face cards, 4 aces. Two dice give 36 outcomes.

Common mistakes

Confusing mutually exclusive with independent. Fix: Exclusive events cannot occur together. If both have non-zero probability, they are in fact dependent, because one occurring rules out the other.
Writing the sample space of two dice as 12 outcomes. Fix: Each die has 6 outcomes and they combine, so n = 6 × 6 = 36.
Counting the favourable outcomes with nCr but the total with a different method, such as nPr or by order. Fix: Choose one method for the whole question. For unordered draws, use nCr for both the numerator and the denominator.
Treating (H, T) and (T, H) as one outcome for two coins, or treating a sum of 7 on two dice as one outcome. Fix: List ordered pairs. Two coins give 4 outcomes. Two dice give 36 ordered pairs, and a sum of 7 has 6 of them.
Adding P(A) and P(B) when the events overlap. Fix: Ask: can both happen together? If yes, subtract P(A ∩ B).
Treating events as mutually exclusive when they are not. Fix: Mutually exclusive means P(A ∩ B) = 0. Test by looking for even one common outcome, such as the king of hearts.
Treating mutually exclusive events as independent. Fix: Mutually exclusive means they cannot occur together, so P(A ∩ B) = 0. Independent means P(A ∩ B) = P(A) × P(B).
Multiplying P(A) × P(B) when the draw is without replacement. Fix: Reduce the total and the favourable count after the first draw, then use P(B | A).
Dividing by the wrong probability, for example computing P(A ∩ B) ÷ P(A) for P(A | B). Fix: The event after the bar is the denominator. Mark the "given" event before you start.
Treating P(A | B) as equal to P(B | A). Fix: The denominators differ. Check which event has been given and divide by its probability.

Exam tips

Questions on this topic are often definition-based. Learn the exact meaning of mutually exclusive, exhaustive, equally likely and independent.
Memorise standard sample space sizes: coin 2, die 6, two dice 36, three coins 8, deck 52.
For 'at least one', go straight to the complement.
If an option is above 1 or below 0, eliminate it immediately.
Know which approach fits: equally likely outcomes means classical, repeated data means statistical, rules for probability means axiomatic.
Questions in this topic are mostly direct counts. Practise the pack of cards facts and the 36-outcome dice grid until they are automatic.
Always check whether the draw is with or without replacement. The words are easy to miss, and they change the answer.
If options include odds such as 3 : 5 and probabilities such as 3/8, read the question for which one is asked. Examiners often include both as options.