Independent events in probability

What are independent events?

In plain words: draw one card from a pack: E is “a spade”, F is “an ace”. P(E) = 1/4, and knowing that the card is an ace leaves P(E|F) = 1/4 too; likewise P(F|E) = 1/13 = P(F) (NCERT Section 13.4, page 417). Such events are independent: knowing one does not change the probability of the other.

Independence means that knowing one event does not change the probability of the other: P(F|E) = P(F) when P(E) > 0, and P(E|F) = P(E) when P(F) > 0 (NCERT Definition 2, page 418). Putting P(F|E) = P(F) into the multiplication rule gives NCERT’s Definition 3, which also works when a probability is 0:

Events that are not independent are dependent. Two experiments are independent when every event of one is independent of every event of the other: tossing a coin and throwing a die, or two throws of a die (page 418; NCERT Example 11, page 419, treats two throws of a die this way).

NCERT Example 10 (page 419): on one die, E = {3, 6} (a multiple of 3) and F = {2, 4, 6} (even) give P(E ∩ F) = 1/6 = (1/3)(1/2), so E and F are independent, though they share the outcome 6.

In the coin and dice examples, assume fair coins, unbiased dice and independent tosses or rolls unless stated otherwise. Card and ball draws are uniformly random from the available objects.

Independent is not mutually exclusive

“Mutually exclusive” is about the events themselves: they have no common outcome. “Independent” is about their probabilities. If P(E) and P(F) are both nonzero, mutually exclusive events cannot be independent, because P(E ∩ F) = 0 while P(E) P(F) > 0; and independent events cannot be mutually exclusive (Remark (ii), page 418).

Three events

A, B and C are mutually independent exactly when all four conditions hold (pages 418–419):

Pairs can be independent while the three are not (extras Q11, beyond the NCERT text), so check all four conditions. NCERT Example 12 (pages 419–420) checks pairs: for three coins, E = “three heads or three tails”, F = “at least two heads” and G = “at most two heads”, the pair E, F is independent, but E, G and F, G are dependent.

Complements of independent events

If E and F are independent, so are E and F′, E′ and F, and E′ and F′ (NCERT Example 13 and Note, pages 420–421). The first follows from P(E ∩ F′) = P(E) − P(E ∩ F) = P(E)(1 − P(F)). A useful consequence (NCERT Example 14, page 421):

Solved examples

NCERT slip (page 422): Exercise 13.2, Q10 gives P(A) = 1/2, P(B) = 7/12 and P(not A or not B) = 1/4. Then P(A ∩ B) = 1 − 1/4 = 3/4, which is larger than P(A) = 1/2: no events can have these probabilities. The key’s answer, “A and B are not independent”, follows from 3/4 ≠ (1/2)(7/12) = 7/24. The correct mathematical conclusion is that the data are inconsistent; no such pair of events exists.

Common mistakes to avoid

Quick revision summary

• E and F are independent if P(E ∩ F) = P(E) P(F), equivalently P(F|E) = P(F) (for P(E) ≠ 0).
• Independent and mutually exclusive are different: with nonzero probabilities, an event pair cannot be both.
• Three events are mutually independent exactly when all three pairs and the triple multiply.
• Complements of independent events are independent, and P(A ∪ B) = 1 − P(A′) P(B′).

Questions students ask

How do you check whether two events are independent?

To check whether two events are independent, find P(E), P(F) and P(E ∩ F) and test whether P(E ∩ F) = P(E) P(F), or, when P(F) ≠ 0, whether P(E|F) = P(E); a feeling that the events are unrelated is not a test. In families of three children, each child independently equally likely to be a boy or a girl, “children of both sexes” (6 of the 8 outcomes) and “at most one girl” (4 of the 8) are independent, because “exactly one girl” has probability 3/8 = (6/8)(4/8). In families of two children the same events are dependent: their probabilities are 1/2 and 3/4, but exactly one girl has probability 1/2, not 3/8.

What is the difference between independent and dependent events?

Two events are independent when P(E ∩ F) = P(E) P(F), so knowing that one happened leaves the other’s probability unchanged, and dependent when this fails, so the information raises or lowers it (NCERT Remark (i), page 418). Dependence is about information, not cause: events about different draws are independent when each draw is freshly random from the same contents with replacement and usually dependent without replacement. For two cards drawn without replacement, the chance that the second card is a king is 1/13, but given that the first card was a queen it is 4/51, slightly more, because a queen has left the pack while all 4 kings remain.

What is the difference between pairwise and mutual independence?

Three events are pairwise independent, a name beyond the NCERT text, when each of the three pairs multiplies, as in P(A ∩ B) = P(A) P(B), and mutually independent when, in addition, P(A ∩ B ∩ C) equals P(A) P(B) P(C) (NCERT Remark (iv), page 418): mutual independence implies pairwise independence, but not the other way round. The triple product alone is not enough either. Throw two dice, with A “the first die shows 1, 2 or 3”, B “the first die shows 3, 4 or 5” and C “the sum is 9”: A ∩ B ∩ C is only (3, 6), with probability 1/36, which equals (1/2)(1/2)(1/9), but P(A ∩ B) = 1/6, not 1/4, so the three are not independent.

How do you find the probability that at least one of three independent events happens?

For three mutually independent events A, B and C, the probability that at least one happens is 1 minus P(A′) P(B′) P(C′), because “none of them” is A′ ∩ B′ ∩ C′ and, beyond the NCERT text, which proves it for two events, complements of mutually independent events are again mutually independent; the probability that all three happen is P(A) P(B) P(C). For three smoke alarms that work independently with probabilities 0.9, 0.8 and 0.7, the chance that none sounds is 0.1 × 0.2 × 0.3, which is 0.006, so at least one sounds with probability 0.994, while all three sound with probability 0.9 × 0.8 × 0.7, which is 0.504. Pairwise independence alone does not justify these products.

Can an event be independent of itself?

An event E is independent of itself exactly when P(E) = 0 or P(E) = 1: by NCERT’s Definition 3 (page 418), independence needs P(E ∩ E) to equal P(E) P(E), and since E ∩ E = E this says P(E) = P(E)^², which holds only for those two values. The sure event S and the impossible event φ are in fact independent of every event, themselves included, because P(E ∩ S) = P(E) × 1 and P(E ∩ φ) = 0 = P(E) × 0. An event with 0 < P(E) < 1, such as “a six” on a die, is not independent of itself: knowing that it happened raises its probability from 1/6 to 1.

Practice questions

Try each question first, then open the answer to check your working.

1. Q1. If P(A) = 3/5 and P(B) = 1/5, find P(A ∩ B) if A and B are independent (NCERT Exercise 13.2, Q1, page 421).

   Show answerHide answer

   P(A ∩ B) = (3/5)(1/5) = 3/25.

2. Q2. P(E) = 3/5, P(F) = 3/10 and P(E ∩ F) = 1/5. Are E and F independent (NCERT Exercise 13.2, Q6, page 421)?

   Show answerHide answer

   P(E) P(F) = 9/50, which is not 1/5 = 10/50, so they are not independent. (Same numbers as NCERT Exercise 13.1, Q1: P(E|F) = 2/3 ≠ P(E) = 3/5 says the same.)

3. Q3. A and B are independent with P(A) = 0.3 and P(B) = 0.4. Find P(A ∩ B), P(A ∪ B), P(A|B) and P(B|A) (NCERT Exercise 13.2, Q8, page 422).

   Show answerHide answer

   0.12; 0.3 + 0.4 − 0.12 = 0.58; and, by independence, P(A|B) = 0.3 and P(B|A) = 0.4.

4. Q4. A die is tossed thrice. Find the probability of getting an odd number at least once (NCERT Exercise 13.2, Q12, page 422).

   Show answerHide answer

   The throws are independent, and P(no odd number) = (1/2)³ = 1/8, so the answer is 7/8.