Probability problems

Most problems follow one recipe: list the sample space, mark the favourable outcomes, and divide. But first check that the outcomes are equally likely.

Not every two-outcome experiment is 50-50. A car starts or it doesn't, but it usually starts. A bag of 3 red and 7 blue marbles gives blue 7/10.

Data gives probabilities too. If 445 of 1000 tyres last more than 14 000 km, the probability is 0.445.

Without replacement, the second draw changes. From 4 red and 5 blue: P(red then blue) = 4/9 × 5/8 = 5/18. After one red has gone, there are only 8 balls left.

'At least one' is often easiest as 1 – P(none). With two coins, P(at least one head) = 1 – P(TT) = 1 – 1/4 = 3/4.

With areas. A die dropped at random on a 3 m × 2 m rectangle lands in a circle of diameter 1 m with probability (π/4)/6 = π/24 ≈ 0.13.

List everything possible and nothing impossible. For the number of heads with three coins, the sample space is {0, 1, 2, 3}. (But these four are not equally likely: there is only one way to get 0 heads and three ways to get 1.)