Applying Bayes' theorem

Routine: name the hypotheses and check they form a partition; write each P(Eᵢ) and each P(A|Eᵢ); multiply in pairs; divide the required product by the sum of all products.

Three boxes hold two coins each: gold-gold, silver-silver, gold-silver. A box is picked at random and a coin taken from it is gold. P(the other coin is gold) = P(box I | gold) = (1/3)(1) / [(1/3)(1) + (1/3)(0) + (1/3)(1/2)] = 2/3.

A test detects HIV in 90% of those who have it and wrongly reports 1% of those who do not. If 0.1% of a large population has HIV, then for a person who tests positive P(has HIV | positive) = (0.001)(0.9) / [(0.001)(0.9) + (0.999)(0.01)] = 90/1089 ≈ 0.083. A rare condition keeps the posterior low even with a good test.

Machines A, B, C make 25%, 35%, 40% of a factory's bolts, with 5%, 4%, 2% of their output defective. For a bolt found defective, P(made by B) = 0.0140/(0.0125 + 0.0140 + 0.0080) = 0.0140/0.0345 = 28/69.

A doctor comes by train, bus, scooter or other transport with probabilities 3/10, 1/5, 1/10, 2/5, and is late with probabilities 1/4, 1/3, 1/12 and 0. Given that he is late, P(train) = (3/40) / (3/40 + 1/15 + 1/120 + 0) = 1/2.

A man speaks the truth 3 times out of 4. He throws a die and reports a six. P(it really is a six) = (1/6)(3/4) / [(1/6)(3/4) + (5/6)(1/4)] = 3/8.

Four boxes of coloured balls. Box I: 3 black, 4 white, 5 red, 6 blue (18 balls). Box II: 2 of each colour (8). Box III: 1 black, 2 white, 3 red, 1 blue (7). Box IV: 4 black, 3 white, 1 red, 5 blue (13). A box is picked at random and a ball drawn from it is black. P(black | box) = 3/18, 2/8, 1/7, 4/13, and P(box III | black) = (1/7)/(3/18 + 2/8 + 1/7 + 4/13) = 156/947 ≈ 0.165.

A correctly set-up machine makes 90% acceptable items, a wrongly set-up one 40%, and 80% of set-ups are correct. After it makes 2 acceptable items, P(correct set-up) = (0.8)(0.9)² / [(0.8)(0.9)² + (0.2)(0.4)²] = 0.648/0.680 = 81/85 ≈ 0.95.