Class 10 › Mathematics › Chapter
Probability
Empirical and theoretical probability, experiments, events, equally likely outcomes — and the four standard random experiments: a coin, a die, two dice, and a deck of cards.
The oldest way of measuring probability is by actually performing an experiment and counting what happens. The empirical (or experimental) probability of an event is worked out from the results of trials that have really been carried out.
For instance, if a coin is tossed $200$ times and a head turns up $108$ times, the empirical probability of getting a head is $\dfrac{108}{200} = 0.54$.
Theoretical probability (also called classical probability) does not need even a single trial to be performed. It is worked out purely by reasoning — by counting how many outcomes are possible in total, and how many of those outcomes favour the event, assuming every outcome is equally likely
The exact formula for theoretical probability is built up in later sections, once an experiment, a random experiment, an event, and equally likely outcomes have all been defined.
An experiment is any operation or activity that produces some well-defined result, called an outcome.
An experiment is called a random experiment when:
- it can be repeated any number of times under identical conditions;
- all of its possible outcomes are known in advance; but
- the exact outcome of any one trial cannot be predicted with certainty before it is actually performed.
These four are used again and again throughout this chapter, so they are worth knowing inside-out. They're set up as standard random experiments in Sections 5–8 below.
A coin has two faces — Head (H) and Tail (T). Tossing one fair (unbiased) coin once gives:
When two coins are tossed together (equivalently, one coin tossed twice), every outcome is a pair — one result per coin:
In general, tossing $n$ coins together gives $2^n$ equally likely outcomes — for three coins, for example, $S = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}$ and $n(S) = 2^3 = 8$.
A die (plural: dice) is a cube whose six faces are marked $1$ to $6$. Throwing one fair (unbiased) die once gives:
When two dice are thrown together, every outcome is an ordered pair $(x, y)$, where $x$ is the number on the first die and $y$ is the number on the second — both from $1$ to $6$.
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | (1,1) | (1,2) | (1,3) | (1,4) | (1,5) | (1,6) |
| 2 | (2,1) | (2,2) | (2,3) | (2,4) | (2,5) | (2,6) |
| 3 | (3,1) | (3,2) | (3,3) | (3,4) | (3,5) | (3,6) |
| 4 | (4,1) | (4,2) | (4,3) | (4,4) | (4,5) | (4,6) |
| 5 | (5,1) | (5,2) | (5,3) | (5,4) | (5,5) | (5,6) |
| 6 | (6,1) | (6,2) | (6,3) | (6,4) | (6,5) | (6,6) |
(1,2) and (2,1) are different ordered pair, hence different outcome
A standard, well-shuffled deck has $52$ cards, split into $4$ suits of $13$ cards each — and, by colour, into $2$ groups of $26$.
- Each suit has $13$ cards: Ace, $2, 3, 4, 5, 6, 7, 8, 9, 10$, Jack, Queen, King.
- Face cards = Jack, Queen, King → $3$ per suit $\times\ 4$ suits = $12$ face cards in the whole deck.
- Aces = $4$ in all, one per suit — an ace is not counted as a face card.
- "Well-shuffled" means every one of the $52$ cards is equally likely to be the one drawn.
The set of all possible outcomes of a random experiment is called its sample space, written $S$. Each outcome inside $S$ is a sample point. For a single die, $S = \{1,2,3,4,5,6\}$.
For example, when a single die is thrown, let $E$ = "the number obtained is even." The outcomes favourable to $E$ are $2$, $4$ and $6$, so $E = \{2, 4, 6\} \subset S$ — exactly what the Venn diagram below shows.
The blue region is the event $E$ — the subset of the sample space $S$ that favours "getting an even number."
The outcomes of an experiment are said to be equally likely when each one has exactly the same chance of occurring as every other outcome — no single outcome is favoured over any other.
Every theoretical-probability calculation in this chapter assumes the outcomes are equally likely — the standard experiments of Sections 5–8 (a fair coin, a fair die, a pair of fair dice, a well-shuffled deck) are always taken to satisfy this unless a question explicitly says a die or coin is biased.
For example, when a single die is thrown once: "getting a number less than $7$" is a sure event (every one of the six faces qualifies), while "getting the number $8$" is an impossible event (no face shows an $8$).
The probability of any event always lies between these two extremes — $0$ for an impossible event and $1$ for a sure event.
For an event $E$, the event "$E$ does not occur" is called the complement of $E$, written $\bar{E}$ (or $E'$, or "not $E$"). It consists of every outcome in $S$ that is not in $E$ — so together, $E$ and $\bar E$ make up the whole sample space, with no overlap between them.
Example: for a single die, if $E$ = "getting an even number" $= \{2,4,6\}$, then $\bar E$ = "getting an odd number" $= \{1,3,5\}$. Here $P(E) = \frac{3}{6} = \frac{1}{2}$ and $P(\bar E) = \frac{3}{6} = \frac{1}{2}$ — and indeed $P(E) + P(\bar E) = 1$.
Bringing everything together: for a random experiment whose outcomes are all equally likely, the (theoretical) probability of an event $E$ is defined as:
- $P(E)$ is never negative and never exceeds $1$: $0 \le P(E) \le 1$.
- $P(S) = 1$ — the sample space itself is a sure event.
- $P(\emptyset) = 0$ — the empty set is an impossible event.
- $P(E) + P(\bar E) = 1$ for every event $E$.
- This formula only applies when every outcome in $S$ is equally likely — that's exactly why the coin, die, dice and card examples always specify "fair" / "unbiased" / "well-shuffled."