Probability for CA Foundation
Measuring chance: classical probability, the addition and multiplication rules, conditional probability and expected value.
Quantitative AptitudePaper 3, Statistics12 MCQs2 lessons2 flashcards3 memory tricks
What you'll learn
- Basic probability and the addition rule
Calculate classical probabilities and use the addition theorem. - Multiplication rule, conditional probability and expectation
Use independence, conditional probability and expected value.
Key formulas
| Classical probability | P(A) = favourable ÷ total equally likely outcomes |
|---|---|
| Range | 0 ≤ P(A) ≤ 1 |
| Complement | P(not A) = 1 − P(A) |
| Addition theorem | P(A ∪ B) = P(A) + P(B) − P(A ∩ B) |
| Odds in favour a : b | P = a/(a + b) |
| Independent events | P(A ∩ B) = P(A) × P(B) |
| Conditional probability | P(A | B) = P(A ∩ B) ÷ P(B) |
| General multiplication | P(A ∩ B) = P(B) × P(A | B) |
Concepts explained
Classical probability
P(A) = favourable outcomes ÷ total equally likely outcomes; it always lies between 0 (impossible) and 1 (certain).
What it means
- Coin: 2 outcomes; two coins: 4; die: 6; two dice: 36; pack: 52 cards.
- P(not A) = 1 − P(A).
- Odds in favour a : b → P = a/(a + b); odds against b : a.
- For 'at least one', use 1 − P(none).
- Sum 7: six favourable pairs out of 36
Answer: 1/6
- Two coins: P(no head) = 1/4
Answer: 3/4
Related terms
- Sample space: the set of all possible outcomes
- Equally likely: no outcome is favoured over another
Exam tip: With two dice, write the 6 × 6 table once in your rough work; it answers most sum questions.
Addition and multiplication rules
'Or' adds probabilities (subtracting any overlap); 'and' multiplies them for independent events.
What it means
- Addition theorem: P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
- Mutually exclusive: P(A ∩ B) = 0, so just add.
- Independent: P(A ∩ B) = P(A) × P(B).
- Without replacement, the second probability changes: P = P(first) × P(second | first).
- P(A) = 0.5, P(B) = 0.4, P(A ∩ B) = 0.2
Answer: P(A ∪ B) = 0.7
- (4/52) × (3/51)
Answer: 1/221
Related terms
- Exhaustive events: together cover the whole sample space
Exam tip: Mutually exclusive and independent are different: mutually exclusive events (with non-zero probability) can't be independent.
Conditional probability and expected value
Conditional probability P(A | B) = P(A ∩ B) ÷ P(B) is the chance of A once we know B has happened. Expected value is the long-run average, Σ x·P(x).
What it means
- Knowing B shrinks the sample space to B.
- If A and B are independent, P(A | B) = P(A).
- E(X) = Σ x·P(x); for a fair die, 3.5.
- Expected value need not be a possible outcome.
- P(A ∩ B) = 0.2, P(B) = 0.5
Answer: P(A | B) = 0.4
- Win ₹100 with probability 0.1, else nothing
Answer: E = ₹10
Related terms
- Bayes' theorem: reverses a conditional probability
Exam tip: Divide by the probability of the condition (the 'given' event), not of the other event.
Memory tricks
P(A ∪ B) = P(A) + P(B) − P(A ∩ B); P(A ∩ B) = P(A)P(B) for independent events.
P(at least one) = 1 − P(none).
Odds a : b in favour → a/(a + b).
Try 3 CA Foundation-style questions
Q1. If P(A) = 0.3, then P(not A) is
- 0.3
- 1.3
- 0
- 0.7
Show answer
Answer: (4) 0.7
P(Aᶜ) = 1 − P(A) = 0.7.
Q2. Two cards are drawn from a pack of 52 without replacement. The probability that both are aces is
- 1/169
- 1/13
- 4/52
- 1/221
Show answer
Answer: (4) 1/221
P = (4/52) × (3/51) = 12/2652 = 1/221.
Q3. If the odds in favour of an event are 3 : 5, its probability is
- 3/5
- 3/8
- 5/8
- 5/3
Show answer
Answer: (2) 3/8
Odds 3 : 5 means 3 favourable for every 5 unfavourable: P = 3 ÷ (3 + 5) = 3/8.
Practise all 12 questions Free · explanations for every option · see where you go wrong