Chapter 14 — Probability
Will the toss go your way? Will it rain on match day? Probability puts a number between 0 and 1 on a chance — and for coins, dice, cards and bags of balls, you can find that number just by counting.
Before a cricket match the captains toss a coin. Nobody can say which side will land up, yet everybody agrees the toss is fair: heads and tails have the same chance. Before a ludo game you hope for a six, knowing that the die has six faces and only one of them is a six. A weather report says there is a "70% chance of rain", and a shopkeeper who draws a lucky coupon from a box knows that most of the coupons are blanks. In each case the result of one try cannot be predicted, but the chance of each result can be described by a number. That number is called a probability.
In this last chapter of the course you meet the simplest and most useful kind of probability, the theoretical (or classical) one. When all the possible results of an experiment are equally likely, the probability of an event is found by counting: the number of results that favour the event divided by the total number of possible results. In Section 14.1 you learn the idea, the words that go with it and two basic rules — a probability always lies between $0$ and $1$, and the probabilities of an event and of "not that event" add up to $1$. In Section 14.2 you use it on the standard settings of board exams: tossing coins, throwing one or two dice, and drawing a card from a pack of 52. The skill that matters most there is careful counting, and the live grids in the labs will do the counting with you.
- list the possible outcomes of an experiment, decide whether they are equally likely, and describe an event as a set of favourable outcomes;
- find the probability of an event as $\dfrac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$ and give it as a fraction in lowest terms, a decimal or a percentage;
- explain why a probability can never be negative or greater than $1$, and recognise impossible events ($P=0$) and sure events ($P=1$);
- use $P(E)+P(\overline{E})=1$ to find the probability of "not $E$" and to shorten "at least one" problems;
- write down the $4$ outcomes for two coins, the $8$ for three coins and the $36$ for two dice, and use them to answer questions about heads, tails, sums and doublets;
- find probabilities for a pack of $52$ playing cards (suits, colours, face cards, kings, aces…), for bags of coloured balls and for spinners with equal sectors.
Sections in this chapter
Probability — A Theoretical Approach
Outcomes, equally likely outcomes and events; the probability of an event as favourable over total; impossible and sure events; complementary events and the rule $P(E)+P(\overline{E})=1$.
Probability with Coins, Dice and Cards
The sample spaces of two and three coins, of one and two dice (the 36-cell grid) and of a pack of 52 cards, with spinners and bags — and how to count favourable outcomes without missing any.