Probability

Data and probability, grade 8 Lesson 11 of 14Learning

Probability

Probability is a number that says how likely something is to happen, when there is no knowing in advance what will happen.

When a die is thrown, you know what can come up – a number from 1 to 6 – but not which number will.

Throw the die a few times. Before each throw, try to guess what will come up.

Experiments and outcomes

An experiment is an action whose result depends on chance: there is no knowing it in advance. Everything that can come up in an experiment is an outcome.

Throwing a die: six outcomes – 1, 2, 3, 4, 5, 6.
Tossing a coin: two outcomes – heads and tails.
Spinning a spinner – a disc divided into parts, with an arrow at its centre that is spun: the outcome is the part the arrow stops on.
Drawing a marble from a bag without looking: each marble is an outcome.
Drawing the name of one pupil in a class of 30: 30 outcomes – each pupil is an outcome.

Events

An event is a set of outcomes of the experiment, described in words. An event happens when the outcome that came up is one of its outcomes.

When a die is thrown, the event “an even number comes up” is the outcomes 2, 4 and 6.
A 4 comes up – the event happened.
A 3 comes up – the event did not happen.
It is handy to call an event by a letter: $\displaystyle A$ – “an even number comes up”.

Equally likely outcomes

When all the outcomes of an experiment have the same chance of coming up, they are equally likely. So it is when a coin is tossed, when a fair die is thrown (no face of it comes up more often than the others) and in a draw without looking. Then the probability of an event is the share of its outcomes out of all the outcomes:

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

$\displaystyle P(A)$ – the probability of the event $\displaystyle A$ ;
$\displaystyle k$ – the number of outcomes in the event $\displaystyle A$ ;
$\displaystyle n$ – the number of all the outcomes of the experiment.

Example: a number greater than 4

The example below finds the probability that a number greater than 4 comes up when a die is thrown.

All the outcomes: 1, 2, 3, 4, 5, 6 – six equally likely outcomes.
The event $\displaystyle A$ – “a number greater than 4 comes up”: the outcomes 5 and 6, two outcomes.

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

Example: a marble from a bag

The example below finds the probability of drawing a blue marble from a bag.

A bag holds 3 red marbles, 5 blue marbles and 2 green marbles.
One marble is drawn without looking.

The three colours do not have the same chance: there are more blue marbles.
The equally likely outcomes are the marbles: 10 outcomes.
The event $\displaystyle B$ – “the marble is blue”: 5 marbles out of the 10.

P(B)=\frac{5}{10}=\frac{1}{2}

Certain and impossible events

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 0 to 1.
An impossible event holds no outcome, and its probability is 0. When a die is thrown: “a 7 comes up”.
A certain event holds all the outcomes, and its probability is 1. When a die is thrown: “a number less than 7 comes up”.
The nearer the probability is to 1, the more likely the event is to happen.

Comparing chances

To tell in which experiment an event has the better chance, compare the probabilities, not the numbers of outcomes.

The example below checks from which bag the chance of drawing a red marble is greater.
Bag A holds 3 red marbles and 2 blue marbles.
Bag B holds 5 red marbles and 5 blue marbles.

In bag A the probability of red is $\displaystyle {\frac{3}{5}}={0.6}$ .
In bag B the probability of red is $\displaystyle {\frac{5}{10}}={0.5}$ .
$\displaystyle 0.6>0.5$ : the chance of drawing a red marble is greater from bag A, although bag B holds more red marbles.
To compare fractions, it is handy to write them as decimals, or with the same denominator.

A spinner with parts of different sizes

The calculation $\displaystyle \frac{k}{n}$ works only when the outcomes are equally likely. The spinner in the picture has four colours, but the blue part is half the disc, so the arrow stops on it more often.

Divide the disc into six equal parts: three blue, one red, one yellow and one green.
The six parts are equally likely, and blue is three of them.
The probability of blue is $\displaystyle {\frac{3}{6}}={\frac{1}{2}}$ , not $\displaystyle \frac{1}{4}$ .
The probability of red is $\displaystyle \frac{1}{6}$ .

Exercise 1

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

An odd 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

A number less than 3 comes up.

/ – 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

A 5 comes up.

/ – 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, 6 blue marbles and 2 yellow marbles.
One marble is drawn without looking.
Find the probability of each event.
Write each probability as a fraction in its lowest terms.

The marble is red.

/ – 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

The marble is yellow.

/ – 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

The marble is red or yellow.

/ – 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

The marble is green.

/ – 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

On the annual school trip, a prize is drawn among the 40 pupils of the grade.
Each pupil gets a ticket with a number of their own, from 1 to 40.
One ticket is drawn without looking.

What is the probability that the number drawn ends in 0?

/ – 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

What is the probability that the number drawn is greater than 30?

/ – 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

Roni has the ticket with the number 17.
What is the probability that Roni wins the prize?

/ – 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

A spinner is divided into eight equal parts, numbered 1 to 8.
For each event, write whether it is certain, impossible, or neither, and what its probability is.

a. A number less than 9 comes up.
b. A 0 comes up.
c. An even number comes up.
d. A two-digit number comes up.

/ – 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

Galit wants to draw a red marble from one of two bags, without looking.
Bag A holds 2 red marbles and 3 blue marbles.
Bag B holds 4 red marbles and 8 blue marbles.
Galit says: “Bag B holds more red marbles, so I should draw from it.”

Is Galit right? Explain with the probabilities.

Exercise 6

On the spinner in the picture, half the disc is red, a quarter blue and a quarter yellow.
Danny says: “The spinner has three colours, so the probability of red is $\displaystyle \frac{1}{3}$ .”

What is Danny’s mistake? What is the probability of red?

Red

/ – 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

What is the probability that the arrow stops on blue or on yellow?

/ – 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

Formulas: data and probability, grade 8

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Relative frequency

The relative frequency of an answer – $\displaystyle f$ its frequency, $\displaystyle n$ the total of all the frequencies
$\displaystyle \frac{f}{n}$

Measures of centre and the range

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
The mean from a frequency table – $\displaystyle f_1, \dots, f_k$ the frequencies of the values $\displaystyle x_1, \dots, x_k$
$\displaystyle \bar{x}=\frac{x_1f_1+x_2f_2+\dots+x_kf_k}{f_1+f_2+\dots+f_k}$
The place of the median in an ordered list of $\displaystyle n$ data
$\displaystyle \frac{n+1}{2}$
The range – the largest value minus the smallest
$\displaystyle R=x_{\max}-x_{\min}$

Probability

Equally likely outcomes – $\displaystyle k$ the outcomes in the event $\displaystyle A$ , $\displaystyle n$ all the outcomes
$\displaystyle P(A)=\frac{k}{n}$
An impossible event – 0, a certain event – 1
$\displaystyle 0\le P(A)\le1$
The complement $\displaystyle \overline{A}$ – $\displaystyle A$ does not happen
$\displaystyle P(\overline{A})$ $\displaystyle =$ $\displaystyle 1-P(A)$
How many times an event $\displaystyle A$ is expected to happen in $\displaystyle n$ experiments
$\displaystyle n\cdot P(A)$