Events and the classical definition of probability | Grade 10

Toss a coin. You cannot say whether it will land heads, but you can say that heads and tails are equally likely. Probability turns this feeling into a number between $0$ and $1$ . The first step is to list carefully what can happen.

Experiments, outcomes and events

A random experiment is a process whose result cannot be predicted with certainty, although all possible results are known (tossing a coin, rolling a die). Each possible result is an outcome. The set of all outcomes is the sample space $\Omega$ .

Classical probability

When the outcomes of an experiment are equally likely, the probability of an event is the fraction of outcomes that favour it.

Two dice

For two dice, an outcome is an ordered pair, so $n(\Omega)=6\cdot6=36$ . The table shows the sum of the two dice for each pair.

1st \ 2nd	1	2	3	4	5	6
1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

Cards and balls

Key terms

English	Tiếng Việt
random experiment	phép thử ngẫu nhiên
outcome	kết quả
sample space	không gian mẫu
event	biến cố
certain event	biến cố chắc chắn
impossible event	biến cố không thể
complement of an event	biến cố đối
probability	xác suất

Summary

$\Omega$ is the set of all outcomes; an event is a subset of $\Omega$ .
For equally likely outcomes, $P(A)=\frac{n(A)}{n(\Omega)}$ .
$0\le P(A)\le 1$ , $P(\Omega)=1$ , $P(\varnothing)=0$ , $P(\bar A)=1-P(A)$ .
"At least one" problems are often easiest through the complement.

Practice

A die is rolled. Find the probability of a prime number.
A coin is tossed twice. Find the probability of at least one tail.
Two dice are rolled. Find the probability that the sum is $8$ .
Two dice are rolled. Find the probability of a double (both numbers equal).
One card is drawn from $52$ . Find the probability that it is a king or a queen.
An urn has $4$ red and $6$ white balls. One ball is drawn. Find the probability that it is white.
A coin is tossed $3$ times. Find the probability of exactly two heads.
Two dice are rolled. Find the probability that the sum is not $7$ .
Challenge. A $3$ -digit number with different digits is formed from $1, 2, 3, 4, 5$ . Find the probability that it is even.
Challenge. A ball is drawn from balls numbered $1$ to $10$ . Find the probability that its number is divisible by $2$ or by $3$ .

Show solutions

Primes: $\{2, 3, 5\}$ , so $\frac36=\frac12$ .
$n(\Omega)=4$ . The complement "no tail" is $\{HH\}$ . $P=1-\frac14=\frac34$ .
Sum $8$ : $(2,6), (3,5), (4,4), (5,3), (6,2)$ gives $5$ outcomes. $P=\frac5{36}$ .
Doubles $(1,1), \dots, (6,6)$ : $6$ outcomes. $P=\frac6{36}=\frac16$ .
$4$ kings and $4$ queens give $8$ cards: $\frac8{52}=\frac2{13}$ .
$\frac6{10}=\frac35$ .
Outcomes $HHT, HTH, THH$ : $\frac38$ .
$P(\text{sum}=7)=\frac16$ , so the answer is $1-\frac16=\frac56$ .
$n(\Omega)=5\cdot4\cdot3=60$ . For an even number the last digit is $2$ or $4$ ( $2$ choices), then $4$ and $3$ choices for the others: $2\cdot4\cdot3=24$ . $P=\frac{24}{60}=\frac25$ .
Divisible by $2$ : $2, 4, 6, 8, 10$ (5 numbers). Divisible by $3$ : $3, 6, 9$ (3 numbers). Together, without counting $6$ twice: $\{2, 3, 4, 6, 8, 9, 10\}$ has $7$ numbers. $P=\frac7{10}$ .