Probability Basics: Coins, Dice and the Bell Curve

How likely is it?

Toss a fair coin. It lands heads or tails, and neither is favoured, so each has probability 1/2. Probability is a number from 0 (impossible) to 1 (certain) that measures how likely something is: the favourable outcomes divided by all the equally likely outcomes. For one throw of a die, the chance of a six is 1/6 ≈ 0.167. Toss two coins and there are four outcomes, HH, HT, TH and TT, so two heads has probability 1/4. A coin has no memory. After five heads in a row the next toss is still 1/2 each way. The belief that tails is 'due' is a very common mistake.

Heads in the long run

Flip a coin ten times and you might see 7 heads, a share of 0.7. That is not odd: the chance of at least 7 heads in 10 flips is (120 + 45 + 10 + 1) / 1024 = 176 / 1024 ≈ 17 %. Flip a thousand times and the share lands much closer to 0.5; a typical distance is about 0.016, so between 0.484 and 0.516. In the drawing the line of the share wobbles and then settles. This is the law of large numbers, proved by Jacob Bernoulli and published in 1713, after his death.

The gap grows, the share settles

A careful point: the law says the share of heads comes close to 1/2, not that heads and tails even out in count. After 100 flips the typical gap between the counts is about 10, and after 10 000 flips it is about 100. The gap grows, but slowly compared with the number of flips, so the share still settles. The South African mathematician John Kerrich, interned in Denmark in World War II, tossed a coin 10 000 times and recorded 5 067 heads, a share of 0.5067. The flips in the drawing are seeded, so you can replay them exactly.

A letter from Pascal

In 1654 a gambler, the Chevalier de Méré, brought Blaise Pascal a puzzle: two players stop a game before the end, so how should they split the prize? Pascal and Pierre de Fermat wrote letters to each other about it, and modern probability grew from that exchange. Try it. The first to win 3 rounds takes 64 coins, and player A leads 2 to 1. At most two rounds remain. B wins only by taking both, with probability 1/2 × 1/2 = 1/4. So A should get 3/4 of the prize, 48 coins, and B 16.

Pascal's triangle

Pascal also studied a triangle of numbers. Start with 1 at the top and make every number the sum of the two above it: 1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1. Row n counts the ways heads can fall in n flips. Row 4 says that among 16 equally likely outcomes of four flips, 1 has no heads, 4 have one, 6 have two, 4 have three and 1 has four. So exactly two heads has probability 6/16 = 0.375. The triangle was known centuries earlier in China, India and Persia, but Pascal linked it to chance.

The Galton board

Sir Francis Galton described a board of pegs in 1889. A ball falls through, and at each peg it bounces left or right with probability 1/2. After 10 rows the ball ends in one of 11 bins. To land in the middle bin it needs exactly 5 rights, which can happen in C(10, 5) = 252 ways out of 1024, so 252 / 1024 ≈ 0.246. An end bin has probability 1/1024. Pour in many balls and the heaps form a bell curve, which Abraham de Moivre had found in 1733. In the sandbox, change the rows and the chance of a right bounce, and the pile slides sideways.

Two dice

Roll two dice and add them. There are 6 × 6 = 36 equally likely pairs. A sum of 2 needs (1, 1), one pair; a sum of 12 needs (6, 6), also one. But a sum of 7 can come as (1, 6), (2, 5), (3, 4), (4, 3), (5, 2) or (6, 1), six pairs. So P(7) = 6/36 = 1/6, the likeliest sum. The counts for sums 2 to 12 are 1, 2, 3, 4, 5, 6, 5, 4, 3, 2, 1, which add up to 36 and draw a little tent, a cousin of the bell curve.

Using the dice

Counting pairs answers many questions. The chance of a sum of at least 10 is (3 + 2 + 1) / 36 = 6/36 = 1/6. The chance of an 8 is 5/36 ≈ 0.139. For two separate events that do not affect each other, multiply: two sixes in a row has probability 1/6 × 1/6 = 1/36. Board games, insurance tables and weather forecasts are all built on such counts. A forecast of 30 % rain means, roughly, that in similar weather it rained on about 3 days out of 10.

Probability and chance