Probability by counting

When outcomes are equally likely, P(E) = n(E)/n(S), and permutations and combinations do the counting. Use ⁿCᵣ when order does not matter (committees, hands of cards) and ⁿPᵣ when it does (finishing orders).

NCERT's Example 8: a committee of two from two men and two women can be formed in ⁴C₂ = 6 ways. P(no man) = 1/6, P(one man) = (2 × 2)/6 = 2/3, P(two men) = 1/6.

NCERT's Example 9: Veena visits four cities A, B, C, D in random order, so n(S) = 4! = 24. A comes before B in 12 of the orders, P = 1/2; A before B before C in 4 orders, P = 4/24 = 1/6.

NCERT's Example 10: a hand of 7 from 52 cards, n(S) = ⁵²C₇. All four kings: ⁴C₄ × ⁴⁸C₃ ways, P = 1/7735. Exactly 3 kings: ⁴C₃ × ⁴⁸C₄ ways, P = 9/1547. At least 3 kings: 9/1547 + 1/7735 = 46/7735.

NCERT's Example 12: five relay teams. Ordered first three places: ⁵P₃ = 5 × 4 × 3 = 60 points. P(A, B, C finish first, second, third in that order) = 1/60; P(A, B and C are the first three in any order) = 3!/60 = 1/10.

Mela volunteers (made up): 2 are picked at random from 5 boys and 3 girls, ⁸C₂ = 28 ways. P(both girls) = ³C₂/28 = 3/28, P(one of each) = (5 × 3)/28 = 15/28, P(at least one girl) = 1 − ⁵C₂/28 = 1 − 10/28 = 9/14.

Mela lucky draw (made up): 3 prize tickets among 20, and a visitor takes 2. P(no prize) = ¹⁷C₂/²⁰C₂ = 136/190, so P(at least one prize) = 54/190 = 27/95.

History: the subject grew out of games of chance. Jerome Cardan (1501–1576) wrote a book on them, published in 1663; in 1654 Pascal and Fermat independently solved a dice problem put to Pascal; Kolmogorov's 1933 book set out the axiomatic theory.