Probability for Beginners

Will it rain tomorrow? Will the coin land heads? One flip tells you nothing. Watch long enough, though, and a pattern surfaces. Probability is the number that puts a price on "unpredictable".

A number between 0 and 1

$P = 0$ : impossible — the sun will not rise in the west;
$P = 1$ : certain — a thrown coin always lands;
$P = \dfrac{1}{2}$ : fifty-fifty — like a coin flip.

The same value can dress three ways: $\dfrac{1}{2} = 0.5 = 50\%$ .

Equally likely outcomes: just count

When every outcome is equally likely, probability turns into a counting game:

$P = \frac{m}{n}$

The numerator $m$ counts the outcomes you want; the denominator $n$ counts all the outcomes there are.

Roll one fair die: the chance of a 6 is $\dfrac{1}{6}$ ; the chance of an even number is $\dfrac{3}{6} = \dfrac{1}{2}$ , since 2, 4 and 6 all count.

Two dice: why 7 is the star

Roll two dice and every combination is equally likely — $6 \times 6 = 36$ outcomes in all. Add each pair of faces and fill in the table (rows: first die, columns: second die):

+	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

Sums of 7 form a diagonal of 6 cells (bold), so $P(7) = \dfrac{6}{36} = \dfrac{1}{6}$ — the champion among all sums. A sum of 2 comes only from $1+1$ , so $P(2) = \dfrac{1}{36}$ , the rarest of all.

The law of large numbers: experiments catch up

Seven heads in ten flips is perfectly normal. But flip ten thousand times and the share of heads closes in on 50% almost every time. The more trials you run, the closer the experimental frequency hugs the theoretical probability. That is the law of large numbers.

InteractiveCoin Flip Lab

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Heads rate

The more you flip, the closer the heads rate gets to 50% — that's the law of large numbers.

Flip 10 times and note the ratio, then 100, then 1000. Watching it wobble and slowly settle is the law of large numbers happening before your eyes.

The coin has no memory

After five heads in a row, the next flip is still $\dfrac{1}{2}$ . The coin owes you no tails — every flip starts fresh.

Check yourself

Quick quiz

1. What is the probability that two dice sum to 2?
2. A bag holds 3 red balls and 1 blue ball. You draw one at random. What is the chance of red?
3. Four flips, four heads. What happens on flip five?