The probability of equally likely outcomes

Probability, grade 9 Lesson 2 of 11Learning

The probability of equally likely outcomes

Probability

The probability of an event is a number that says how likely it is to happen.

The probability of the event $\displaystyle A$ is written $\displaystyle P(A)$ (P for Probability).
A large probability – the event happens often; a small one – rarely.

Equally likely outcomes

When all the outcomes of an experiment have the same chance of coming up, they are equally likely. So it is with a fair die or a coin, and with a draw made without looking. Then an event’s probability is the share of its outcomes among all the outcomes:

\displaystyle P(A)=\frac{k}{n}

$\displaystyle k$ – the number of outcomes in the event $\displaystyle A$ ;
$\displaystyle n$ – the number of outcomes in the sample space.

Example: a number greater than 4

The example below calculates the probability that a die comes up with a number greater than 4.

The sample space: $\displaystyle \{1,2,3,4,5,6\}$ – six equally likely outcomes.
The event $\displaystyle A$ – “a number greater than 4”: $\displaystyle \{5,6\}$ , two outcomes.

P(A)=\frac{2}{6}=\frac{1}{3}

Probability: a number from 0 to 1

The probability of an event is always a number from 0 to 1:

\displaystyle 0\le P(A)\le1

An event holds from 0 to $\displaystyle n$ outcomes, so $\displaystyle \frac{k}{n}$ is from $\displaystyle 0$ to $\displaystyle 1$ .
An impossible event: no outcomes, and its probability is $\displaystyle 0$ .
A certain event: all the outcomes, and its probability is $\displaystyle 1$ .

Outcomes with different chances

The calculation above fits equally likely outcomes only. The spinner in the picture has three outcomes – red, blue and yellow – but the red part is half the circle, so the arrow stops on it more often.

Split the circle into four equal parts: two red, one blue and one yellow.
The four parts are equally likely, and red is two of them.
The probability of red is $\displaystyle {\frac{2}{4}}={\frac{1}{2}}$ , not $\displaystyle \frac{1}{3}$ .

Example: marbles in a bag

The example below calculates the probability of drawing a red marble from a bag holding marbles of two colours.

A bag holds 3 red marbles and 2 blue marbles.
One marble is drawn without looking.

The two colours do not have the same chance: there are more red marbles.
The equally likely outcomes are the marbles: five outcomes.
The event $\displaystyle A$ – “the marble is red”: three marbles out of the five.

P(A)=\frac{3}{5}

Exercise 1

A die is thrown.
Calculate the probability of each event, and simplify the fraction if you can.

a

A 5 comes up.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

A number less than 3 comes up.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

A prime number comes up.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

A bag holds 4 red marbles, 5 blue marbles and 3 green marbles.
One marble is drawn without looking.
Calculate the probability of each event.
Write each probability as a fraction in its lowest terms.

a

The marble is red.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

The marble is red or green.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

The marble is white.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

A class has 30 pupils, and 18 of them are girls.
One pupil is drawn to represent the class on the student council.
Write each probability as a fraction in its lowest terms.

a

What is the probability that a girl is drawn?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

What is the probability that a boy is drawn?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

Noa spins the spinner that is half red, a quarter blue and a quarter yellow.
Noa says: “There are three outcomes, so the probability of blue is $\displaystyle \frac{1}{3}$ .”
What is Noa’s mistake? What is the probability of blue?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

Is there an event whose probability is $\displaystyle \frac{5}{4}$ ? Explain.

Formulas: probability, grade 9

The probability of an event

Equally likely outcomes – $\displaystyle k$ the number of outcomes in the event $\displaystyle A$ , $\displaystyle n$ the number of all outcomes
$\displaystyle P(A)=\frac{k}{n}$
A probability is a number from 0 to 1 – a certain event 1, an impossible event 0
$\displaystyle 0\le P(A)\le1$
The complement $\displaystyle \overline{A}$ – $\displaystyle A$ does not happen
$\displaystyle P(\overline{A})$ $\displaystyle =$ $\displaystyle 1-P(A)$

Relative frequency

The relative frequency – $\displaystyle k$ the number of times the event happened, $\displaystyle n$ the number of times the experiment was done
$\displaystyle \frac{k}{n}$

Two-stage experiments

An outcome table whose cells are equally likely – $\displaystyle k$ the cells in the event $\displaystyle A$ , $\displaystyle n$ all the cells
$\displaystyle P(A)=\frac{k}{n}$
A path on a tree diagram – the product of the probabilities $\displaystyle p_1$ , $\displaystyle p_2$ on its branches; in the area model, the rectangle’s area
$\displaystyle P=p_1\cdot p_2$
An event of several paths – the sum of their probabilities
$\displaystyle P(A)=P_1+P_2+\ldots$