Where do probability numbers come from? When all the outcomes are equally likely, like the faces of a fair number cube, you can work the probability out by counting. It is a number on the probability scale.
Key idea
If all outcomes are equally likely, $$P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of outcomes}}$$
Why it works
Equally likely outcomes share the total probability, $1$, equally. With 6 faces, each face gets $\frac{1}{6}$. An event that includes 3 of the faces gets 3 of those shares: $\frac{3}{6}$. This only works when the outcomes really are equally likely.
Worked examples
Worked example: A number cube
What is the probability of rolling an even number on a fair number cube?
- Outcomes: 1, 2, 3, 4, 5, 6, all equally likely.
- Favorable: 2, 4, 6. That is 3 of 6.
- $P = \frac{3}{6} = \frac{1}{2}$.
Answer: $\frac{1}{2}$
Worked example: A bag of marbles
A bag has 3 red, 5 blue and 2 green marbles. What is the probability of picking blue?
- There are $3 + 5 + 2 = 10$ marbles, each equally likely.
- $P(\text{blue}) = \frac{5}{10} = \frac{1}{2}$.
Answer: $\frac{1}{2}$
Worked example: Numbered cards
Cards numbered 1 to 20 are shuffled. What is the probability of drawing a multiple of 3?
- Multiples of 3: 3, 6, 9, 12, 15, 18. That is 6 cards.
- $P = \frac{6}{20} = \frac{3}{10}$.
Answer: $\frac{3}{10}$
Common mistake
Thinking any event with two outcomes has probability $\frac{1}{2}$: "a lottery ticket wins or loses, so my chance is $\frac{1}{2}$." Favorable over total works only for equally likely outcomes, and winning is far less likely than losing. Ask: are these outcomes really equally likely?