Probability of events

Probability gives each event a number from 0 to 1 that measures how likely it is. The modern definition rests on three rules (the axioms):

• P(E) ≥ 0 for every event, and P(S) = 1.
• If E and F are mutually exclusive, P(E ∪ F) = P(E) + P(F).
• From these, P(φ) = 0, and every probability lies between 0 and 1. An impossible event has probability 0, a sure event 1.

Assigning probabilities to outcomes. Each outcome gets a number from 0 to 1, and they must add up to 1. For four outcomes, 0.1, 0.2, 0.3, 0.4 is a valid assignment; 0.5, 0.5, 0.1, –0.1 is not (a negative number). The probability of an event is the sum over its outcomes.

Equally likely outcomes: P(E) = n(E)/n(S), favourable outcomes over all outcomes. On a fair die, P(more than 4) = 2/6 = 1/3.

If the outcomes are not equally likely, count faces carefully. A die has faces 1, 2, 2, 3, 3, 3: P(3) = 3/6 = 1/2 and P(1 or 2) = 3/6 = 1/2.

The addition rule: P(A ∪ B) = P(A) + P(B) – P(A ∩ B): add, then take away the overlap you counted twice.

• One card from 52: P(heart or face card) = 13/52 + 12/52 – 3/52 = 22/52 = 11/26 (three face cards are hearts).
• For mutually exclusive events the overlap is 0, so just add: with two dice, P(sum 2 or sum 12) = 1/36 + 1/36 = 1/18.

The complement rule: P(not A) = 1 – P(A). It is the quick way to handle 'at least one'. Three coins: P(at least one head) = 1 – P(no head) = 1 – 1/8 = 7/8.

A survey: 50% of a town reads an English paper, 35% a Hindi paper, 20% both. For a person chosen at random:

• P(at least one) = 0.50 + 0.35 – 0.20 = 0.65.
• P(neither) = 1 – 0.65 = 0.35, since 'neither A nor B' is the complement of 'A or B'.
• P(English only) = P(A) – P(A ∩ B) = 0.30.

Working backwards: P(pass test A) = 0.7, P(pass B) = 0.6, P(pass at least one) = 0.85. Then P(both) = 0.7 + 0.6 – 0.85 = 0.45, and P(exactly one) = P(A) + P(B) – 2P(A ∩ B) = 0.4.

A small game. Toss a coin three times; win ₹2 for each head and lose ₹1 for each tail. The possible results are 3 heads (+₹6), 2 heads (+₹3), 1 head (₹0) and 0 heads (–₹3), with probabilities 1/8, 3/8, 3/8 and 1/8. Count the outcomes in S for each case; the four probabilities add up to 1.