Mathematics · Combinatorics, probability and statistics · Grade 9

Probability: coin, die, Venn

If we toss a coin 1000 times, how many heads will we get? What is the probability that two dice differ by 1?

The situation

The weather forecast says “30% chance of rain”, a lottery ticket says “every tenth ticket wins”, and in a game a die is rolled. If we toss a coin 1000 times, how many heads will come up — exactly 500? When two dice are rolled, which total comes up most often? Chance can be counted: we divide the favourable outcomes by all the outcomes, repeat the experiment many times, and link events with “or”, “and”, “not”.

Topic

Events and probability: the classical definition, relative frequency, operations on events, the addition and multiplication rules. A coin, a die and a lottery: over a thousand throws the frequency approaches the probability; “or” and “and” with Euler–Venn circles

Key ideas

Classical definition
Relative frequency
Operations on events

Formulas

P (A) = \frac{m}{n}

Classical definition: out of $n$ equally likely outcomes, $m$ are favourable to the event. An even number on a die: $\frac{3}{6} = \frac{1}{2}$ .

w (A) = \frac{k}{N} \to P (A)

Relative frequency: in $N$ trials the event occurred $k$ times; for large $N$ the frequency approaches the probability (the statistical definition).

P (A \cup B) = P (A) + P (B) - P (A \cap B), P (\overline{A}) = 1 - P (A)

Addition theorem: “A or B”; for mutually exclusive events $P (A \cap B) = 0$ and the probabilities are simply added. The complementary event is the remaining part.

P (A \cap B) = P (A) \cdot P (B) (independent), P (A \cap B) = P (A) \cdot P (B | A)

Multiplication rule: “A and B”. For independent events (two coins) the probabilities are multiplied; when drawing without replacement, the second factor is a conditional probability.

P (A) = \frac{S (A)}{S (Ω)}

Geometric definition: when there are infinitely many outcomes (a point on a target), the probability is a ratio of areas.

Theory

Experiment, outcome, event

Rolling a die is an experiment, the numbers 1 to 6 are its outcomes; if the die is fair, they are equally likely. An event is a part of the outcomes: “an even number comes up” is {2; 4; 6}. An event that happens in every trial is certain (“a number not greater than 6”), one that never happens is impossible (“7 comes up”), and the rest are random. On the first step, choose events for the coin and the die.

Classical definition

If the outcomes are equally likely, the probability is the ratio of the number of favourable outcomes to the number of all outcomes: $P (A) = \frac{m}{n}$ . An even number on a die: $\frac{3}{6} = \frac{1}{2}$ . When two dice are rolled, the outcomes are 36 pairs (a table); “sum 7” occupies 6 cells — $\frac{6}{36} = \frac{1}{6}$ , while “sum 2” occupies only one — $\frac{1}{36}$ . A probability is always between 0 and 1: an impossible event has 0, a certain one has 1.

Relative frequency and the statistical definition

If we toss a coin 20 times and get 12 heads, the relative frequency of heads is $\frac{12}{20} = 0.6$ — not the probability $0.5$ , but close. As the number of trials $N$ grows, the frequency fluctuates less and less around the probability (the law of large numbers). That is why, when outcomes are not equally likely (a bent coin, rain, a patient recovering), the probability is defined statistically: as the frequency in many trials. On the third step, change the length of the series.

Three definitions of probability

Classical — equally likely outcomes are counted; statistical — the frequency is observed; geometric — there are infinitely many outcomes (a point hitting a target), and the probability is a ratio of areas: for a circle of radius 3 on a square target of side 10, the probability of hitting the circle is $\frac{π \cdot 9}{100} \approx 0.28$ . On the fourth step random points are “shot”: the share that land in the circle approaches this number — this is the Monte Carlo method.

Operations on events

$A$ — “plays football”, $B$ — “plays chess”. The union $A \cup B$ — at least one of them (or), the intersection $A \cap B$ — both (and), the complement $\overline{A}$ — does not play football. In an Euler–Venn diagram these are the union of the circles, their overlap, and the outside of the circle. Addition theorem: $P (A \cup B) = P (A) + P (B) - P (A \cap B)$ — the overlap has been added twice, so it is subtracted once; if the events are mutually exclusive, the overlap is empty and the probabilities are simply added. $P (\overline{A}) = 1 - P (A)$ .

The multiplication rule and a note on the model

Two coins are independent events: the result of the second does not depend on the first, $P (H H) = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}$ . When a ball is drawn from a box without replacement, the second draw depends on the first: both red from 3 red and 2 blue — $\frac{3}{5} \cdot \frac{2}{4}$ ; the second factor is the conditional probability $P (B | A)$ . In a tree diagram the probabilities along the branches are multiplied, and those of the leaves are added. The coin and die in the experiment are ideal (equally likely outcomes); the “random” series are produced on the computer by a seeded generator, so the “New series” button gives a different result, while replaying shows the same one.

Did you know?

The theory of probability began in 1654 with the correspondence of Blaise Pascal and Pierre de Fermat: they solved the problem of “how to divide the stakes in an unfinished game”, posed by the gambler de Méré.
The statistician Karl Pearson tossed a coin 24 000 times and got 12 012 heads — a frequency of 0.5005: very close to the probability 0.5, but not exactly equal.

Inside the experiment

The topic is explained step by step with animation, from simple to advanced: from a real-life picture to the formula and the proof. Then you drag the points yourself to check. Inside are missions, a quiz and a guided tour.

More experiments in this section

Combinatorics: counting rules — Grade 7
Statistics: mean, median, σ — Grade 8