Probability Basics

What is probability?

Probability measures how likely an event is to happen. We write it as a number from 0 to 1:

• 0 means the event is impossible.
• 1 means the event is certain.
• 0.5 means there is an even chance (just as likely to happen as not).

You can write probabilities as fractions, decimals or percentages. A 1/2 chance, 0.5 and 50% all mean the same thing.

The probability scale

Probability	Likelihood	Example
0	Impossible	Rolling a 7 on a normal die
0.25	Unlikely	Drawing a spade from a deck
0.5	Even chance	A fair coin landing heads
0.75	Likely	Drawing a non-spade from a deck
1	Certain	The sun rising tomorrow

Theoretical probability

When all outcomes are equally likely, we can calculate probability directly:

P(event) = number of favourable outcomes ÷ total number of possible outcomes

Worked example: rolling a die. A fair six-sided die has outcomes 1, 2, 3, 4, 5, 6.

• P(rolling a 4) = 1 favourable ÷ 6 total = 1/6
• P(rolling an even number) = 3 favourable (2, 4, 6) ÷ 6 = 3/6 = 1/2

Worked example: a deck of cards. A standard deck has 52 cards, 26 red and 26 black.

• P(red card) = 26/52 = 1/2
• P(a king) = 4/52 = 1/13

Experimental probability

Sometimes outcomes are not equally likely, or we just want to test reality. Then we run trials and use:

Experimental probability = number of times the event happened ÷ total number of trials

If you flip a coin 50 times and get 27 heads, the experimental probability of heads is 27/50 = 0.54. The theoretical value is 0.5. The more trials you run, the closer experimental results usually get to the theoretical value — this is the law of large numbers.

Complementary events

The complement of an event is it not happening. Because something either happens or it does not, these two probabilities always add to 1:

P(event) + P(not event) = 1

Example: if the probability of rain is P(rain) = 0.3, then:

• P(no rain) = 1 − 0.3 = 0.7

This is a handy shortcut: sometimes it is far easier to find the probability of something not happening and subtract from 1.

Outcomes always sum to 1

For any situation, the probabilities of all possible outcomes add up to 1. On a die:

P(1) + P(2) + P(3) + P(4) + P(5) + P(6) = 6 × (1/6) = 1

If your probabilities do not total 1, something has gone wrong.

Combining simple events

For two independent events (where one does not affect the other), multiply their probabilities to find the chance of both happening.

Example: flipping two heads in a row.

• P(head) = 1/2 each time
• P(head AND head) = 1/2 × 1/2 = 1/4

Example: rolling a 6 twice.

• P(6 and 6) = 1/6 × 1/6 = 1/36

Why probability matters

Probability powers weather forecasts, games, insurance, medicine and machine learning. Once you are comfortable counting outcomes and using the 0-to-1 scale, you can reason clearly about chance. Sharpen the fraction skills behind these calculations with introduction to fractions, and the percentage conversions in your number toolkit.