Probability • Topic 3 of 4

Independent Events

Two events are independent when one happening tells you nothing about the other, so P(A|B) = P(A). The test is multiplicative: A and B are independent exactly when P(A∩B) = P(A)·P(B). This is the rule for anything done "with replacement" or for genuinely separate trials — two coin tosses, two rolls of a die, drawing-and-returning a ball. Do not confuse independent with mutually exclusive: mutually exclusive events cannot occur together (overlap zero), whereas independent events usually can. For a chain of independent events, just multiply the individual probabilities. The CAT favourite is "at least one success in n trials", best solved by the complement: P(at least one) = 1 − P(no success)^n. For example, the chance of at least one six in two rolls of a die is 1 − (5/6)² = 11/36.

✅ Solved examples

1. A coin is tossed twice. Find the probability of getting two heads.
Independent tosses: P(H)·P(H) = (1/2)(1/2) = 1/4.
2. A die is rolled twice. Find the probability of getting a 6 on both rolls.
Independent: (1/6)(1/6) = 1/36.
3. The probability A solves a problem is 1/3 and B solves it is 1/4, independently. Find the probability the problem is solved.
P(solved) = 1 − P(neither) = 1 − (2/3)(3/4) = 1 − 1/2 = 1/2.
4. A ball is drawn from a bag of 3 red and 2 green, noted and replaced, then a second ball is drawn. Find the probability both are red.
With replacement the draws are independent: (3/5)(3/5) = 9/25.

✏️ Practice — try these, take hints as needed

1. A coin is tossed 3 times. Probability of three tails?
Each toss independent.
(1/2) three times.
Multiply.
1/8
2. A die is rolled twice. Probability of an even number both times?
P(even) = 1/2.
Independent.
(1/2)(1/2).
1/4
3. P(A) = 0.6, P(B) = 0.5, independent. Find P(A∩B).
Independent ⇒ multiply.
0.6 × 0.5.
As a fraction.
0.3 (3/10)
4. A die is rolled twice. Probability of at least one 6?
Use the complement.
P(no 6) = 5/6 each.
1 − (5/6)².
11/36
5. A and B hit a target with probabilities 1/2 and 1/3 independently. Probability the target is hit?
Complement: neither hits.
(1/2)(2/3) = 1/3.
1 − 1/3.
2/3

📝 Topic test — 8 questions

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