Mathematics from zero
What is probability
Flip a coin. Heads or tails — you cannot know which. But you can say something exact: heads will come up about half the time. “About half” is a number, and putting a number on how likely something is is what probability does.
After this lesson you can say what probability measures, read the 0-to-1 scale, compute the probability of an event with equally likely outcomes, and explain what 0 and 1 mean.
Probability is a number that measures how likely an event is. An event is something that may or may not happen — a coin landing heads, a die showing 3. Its probability is a single number capturing how likely it is. Every probability sits between 0 and 1.
On the 0-to-1 scale, 0 means impossible and 1 means certain. A probability of 0
is an event that cannot happen; a probability of 1 is one that must happen. Halfway
along, 1/2, is an event equally likely to happen or not — like a fair coin landing
heads. The closer to 1, the more likely.
For equally likely outcomes, probability is favourable outcomes over total outcomes. When every outcome is just as likely as every other, the probability of an event is a fraction: how many outcomes count as the event (the favourable outcomes), divided by how many outcomes there are in total.
A die makes this concrete. A fair six-sided die has 6 equally likely outcomes. The
event “roll a 3” has 1 favourable outcome, so its probability is 1/6. The event
“roll an even number” has 3 favourable outcomes — 2, 4, 6 — so its probability is
3/6, which is 1/2.
A bag holds 10 balls, all the same size: 3 are red, 7 are blue. You draw one without looking. What is the probability it is red?
Every ball is equally likely to be drawn, so probability is favourable over total.
Total outcomes: 10 balls could be drawn. Favourable outcomes: 3 of them are red.
So the probability of red is 3/10. As a decimal that is 0.3 — a little less than
half, which fits: red balls are the minority in the bag.
Why this works
Why must a probability stay between 0 and 1? Because it is favourable outcomes divided by total outcomes, and the favourable ones are always part of the total. A part divided by the whole cannot exceed 1, and cannot drop below 0. Probability 1 means every outcome is favourable; probability 0 means none is.
Common mistake
A common mistake is dividing by the favourable count instead of the total — computing
the chance of red as 3/3 = 1. The denominator must be every possible outcome, not
just the favourable ones. Probability is favourable over total: always count the
whole set of outcomes for the bottom number.
What is the probability of an event that is certain to happen? Type the value.
What is the probability of an event that is impossible? Type the value.
Rolling a fair six-sided die, how many outcomes are favourable for 'roll a 3'? Type the count.
Rolling a fair six-sided die, how many outcomes are there in total? Type the count.
A bag has 10 balls, 4 of them green. What is the probability of drawing green, as a decimal? Type it.
For equally likely outcomes, how do you compute the probability of an event?
Probability is a number from 0 to 1 measuring how likely an event is: 0 is impossible, 1 is certain, 1/2 is equally likely either way. When all outcomes are equally likely, the probability of an event is its favourable outcomes divided by the total number of outcomes. Because the favourable outcomes are part of the total, a probability always stays between 0 and 1.