Probability Basics: And, Or, Not and Dice

Probability measures how likely something is, from 0 (impossible) to 1 (certain). For equally likely outcomes, it is favourable outcomes ÷ total outcomes. Three rules cover most problems: not ($1 - P$), and for independent events (multiply), and or (add, then subtract the overlap).

The basic formula

Rolling a 4 on a fair die: $\frac{1}{6}$.

"Not": the complement

Not rolling a 4: $1 - \frac{1}{6} = \frac{5}{6}$.

"And": independent events

If one event does not affect the other, multiply:

A coin lands heads and a die shows 6: $\frac{1}{2} \times \frac{1}{6} = \frac{1}{12}$.

"Or": the addition rule

Subtract the overlap so it is not counted twice. If the events are mutually exclusive (cannot both happen), the overlap is 0.

Two dice

There are $6 \times 6 = 36$ equally likely outcomes. Six of them add to 7 (1+6, 2+5, 3+4, 4+3, 5+2, 6+1), so

7 is the most likely total; 2 and 12 are the least likely ($\frac{1}{36}$ each).

Expected value

The average result over many trials. For a fair die: $\frac{1 + 2 + 3 + 4 + 5 + 6}{6} = 3.5$.

Common mistakes

• Adding probabilities for "and".
• Forgetting to subtract the overlap for "or".
• Assuming past results change independent events (the gambler's fallacy).

Practice questions

1. Probability of drawing a heart from a standard deck?
2. Two coins: probability of two heads?
3. Probability of an even number or a number greater than 4 on a die?

Answers: 1) $\frac{1}{4}$ 2) $\frac{1}{4}$ 3) $\frac{2}{3}$

Combine events with the probability calculator, work out dice totals with the dice probability calculator, and find long-run averages with the expected value calculator. Counting methods are in permutations vs combinations.

Frequently asked questions

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