Read it again any time, including while you work through the exercises.
Rolling a die, drawing a card, flipping a coin: in all these situations, we know the possible results but we cannot predict which one will happen. This is called a random experiment. Probability is the mathematician's tool for measuring the chance that a result occurs.
Some precise vocabulary. Each possible result of an experiment is called an outcome: for a six-sided die, the outcomes are 1, 2, 3, 4, 5 and 6. An event is a condition that groups one or more outcomes: 'getting an even number' groups the outcomes 2, 4 and 6.
The probability of an event is a number ALWAYS between 0 and 1. It is a scale: at 0, the event is impossible (rolling a 7 with a standard die); at 1, it is certain (rolling a number between 1 and 6). In between, the closer the probability is to 1, the more likely the event. At 0.5, exactly in the middle, the event is as likely to happen as not, like heads or tails with a fair coin.
This scale gives you a checking superpower: if a calculation gives you a probability of 1.5 or -0.2, it must be wrong! No probability can leave the interval from 0 to 1. And beware: very likely does not mean certain. A probability of 0.8 is a lot, but the event can still fail to happen.
Place these three events on the probability scale: A 'getting heads with a fair coin', B 'rolling a 7 with a six-sided die', C 'drawing a red ball from a bag that contains only red balls'.
- 1
P(B) = 0
A six-sided die has no 7 on it: event B is impossible, its probability is 0, at the very bottom of the scale.
- 2
P(C) = 1
All the balls are red: event C is certain, its probability is 1, at the very top of the scale.
- 3
P(A) = 0.5
Heads and tails are equally likely: the probability of A is 0.5, exactly in the middle of the scale.
Result: On the scale: B at 0 (impossible), A at 0.5 (one chance in two), C at 1 (certain).