We can make predictions for trials by first creating the sample space and then determining the theoretical probability of each outcome.
The theoretical probability of an event, $E$E, is given by:
|
Probability |
$=$= | $\frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$Number of favourable outcomesTotal number of possible outcomes |
Exploration
If you roll two six-sided dice and add the numbers together, what is the probability of getting a sum of $6$6? What about a sum of $10$10 or greater? $8$8 or less?
Before we answer these questions we need to determine the sample space. The possible outcomes for two dice can be drawn in a grid:
| Second die | |||||||
| $1$1 | $2$2 | $3$3 | $4$4 | $5$5 | $6$6 | ||
| First die | $1$1 | $1,1$1,1 | $1,2$1,2 | $1,3$1,3 | $1,4$1,4 | $1,5$1,5 | $1,6$1,6 |
| $2$2 | $2,1$2,1 | $2,2$2,2 | $2,3$2,3 | $2,4$2,4 | $2,5$2,5 | $2,6$2,6 | |
| $3$3 | $3,1$3,1 | $3,2$3,2 | $3,3$3,3 | $3,4$3,4 | $3,5$3,5 | $3,6$3,6 | |
| $4$4 | $4,1$4,1 | $4,2$4,2 | $4,3$4,3 | $4,4$4,4 | $4,5$4,5 | $4,6$4,6 | |
| $5$5 | $5,1$5,1 | $5,2$5,2 | $5,3$5,3 | $5,4$5,4 | $5,5$5,5 | $5,6$5,6 | |
| $6$6 | $6,1$6,1 | $6,2$6,2 | $6,3$6,3 | $6,4$6,4 | $6,5$6,5 | $6,6$6,6 | |
We can now tell that there are $36$36 possible outcomes. Depending on the trial we can highlight the favourable outcomes corresponding to the event, and the probability of any particular event is given by the formula
$\text{Probability}=\frac{\text{Number of favourable outcomes}}{36}$Probability=Number of favourable outcomes36
Explore this applet to find the various probabilities:
Once we have a sample space with every outcome being equally likely, we can express the probability as a fraction, decimal, or percentage.
Repeating trials
Once we know the probability of an event, we can predict how many times this event will occur if a trial is repeated several times.
We multiply the probability of the event by the number of trials, rounding to the nearest whole number.
Worked example
example 1
A letter is chosen at random from the word "MATHEMATICS" two hundred times. and the results written down in a list.
How many times can we expect that a "T" will be chosen? Round your answer to the nearest whole number.
Think: There are $11$11 letters in the word "MATHEMATICS", and each one is equally likely to be chosen. We can think of the sample space like this spinner:
 |
Once we know the probability of "T" in one trial, we can multiply it by $200$200 to find the expected number of times that "T" will appear in the list.
Do: Using the formula,
$\text{Probability}=\frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}=\frac{2}{11}$Probability=Number of favourable outcomesTotal number of outcomes=211
This means each time a letter is chosen, there is a $\frac{2}{11}$211 chance of choosing a "T".
Multiplying this probability by the number of trials ($200$200) tells us the expected number of times that "T" will be chosen:
$\text{Number of Ts}=200\times\frac{2}{11}=\frac{400}{11}$Number of Ts=200×211=40011
As a decimal this is $36.\overline{36}$36.36, which rounds to the nearest whole number $36$36. This is how many times we should expect "T" to be chosen in the list of $200$200 letters.
Unless the question tells you to round your answer, you should not round it.
The exact value of theoretical probability is often very important, so we won't approximate it with a percentage or decimal unless we are told to do so.
Practice questions
Question 1
A two-digit number is formed using the numbers $3$3 and $2$2. It can be two of the same or one of each number in any order.
What is the probability that the number formed is odd?
What is the probability that the number formed is more than $30$30?
Question 2
An eight-sided die is rolled $25$25 times. How many times should we expect to roll a $7$7? Round your answer to the nearest whole number.
 |
| eight-sided die |
Question 3
A bag contains $28$28 red marbles, $27$27 blue marbles, and $26$26 black marbles.
What is the probability of drawing a blue marble?
A single trial is drawing a marble from the bag, writing down the colour, and putting it back. If this trial is repeated $400$400 times, how many blue marbles should you expect?
Round your answer to the nearest whole number.
In order to predict the future, we sometimes need to determine the probability by running experiments, or looking at data that has already been collected. This is called experimental probability, since we determine the probability of each outcome by looking at past events.
Exploration
Imagine we have a "loaded" die, where a weight is placed inside the die opposite the face that the cheater wants to come up the most (in this case, the $6$6):
If the die is made like this, the probability of each outcome is no longer equal, and we cannot say that the probability of rolling any particular face is $\frac{1}{6}$16.
Instead we will need to roll the die many times and record our results, and use these results to predict the future. Here are the results of an experiment where the die was rolled $200$200 times:
| Result | Number of rolls |
|---|---|
| $1$1 | $11$11 |
| $2$2 | $19$19 |
| $3$3 | $18$18 |
| $4$4 | $18$18 |
| $5$5 | $20$20 |
| $6$6 | $114$114 |
We can now try to predict the future using this experimental data, and the following formula:
$\text{Experimental probability of event}=\frac{\text{Number of times event occurred in experiments}}{\text{Total number of experiments}}$Experimental probability of event=Number of times event occurred in experimentsTotal number of experiments
Here is the table again, with the experimental probability of each face listed as a percentage:
| Result | Number of rolls | Experimental probability |
|---|---|---|
| $1$1 | $11$11 | $5.5%$5.5% |
| $2$2 | $19$19 | $9.5%$9.5% |
| $3$3 | $18$18 | $9%$9% |
| $4$4 | $18$18 | $9%$9% |
| $5$5 | $20$20 | $10%$10% |
| $6$6 | $114$114 | $57%$57% |
A normal die has around $17%$17% chance of rolling a $6$6, but this die rolls a $6$6 more than half the time!
Sometimes our "experiments" involve looking at historical data instead. For example, we can't run hundreds of Eurovision Song Contests to test out who would win, so instead we look at past performance when trying to predict the future. The following table shows the winner of the Eurovision Song Contest from 1999 to 2018:
| Year | Winning country | Year | Winning country |
|---|---|---|---|
| 1999 | Sweden | 2009 | Norway |
| 2000 | Denmark | 2010 | Germany |
| 2001 | Estonia | 2011 | Azerbaijan |
| 2002 | Latvia | 2012 | Sweden |
| 2003 | Turkey | 2013 | Denmark |
| 2004 | Ukraine | 2014 | Austria |
| 2005 | Greece | 2015 | Sweden |
| 2006 | Finland | 2016 | Ukraine |
| 2007 | Serbia | 2017 | Portugal |
| 2008 | Russia | 2018 | Israel |
What is the experimental probability that Sweden will win the next Eurovision Song Contest?
We think of each contest as an "experiment", and there are $20$20 in total. The winning country is the event, and we can tell that $3$3 of the contests were won by Sweden. So using the same formula as above,
$\text{Experimental probability of event}=\frac{\text{Number of times event occurred in experiments}}{\text{Total number of experiments}}$Experimental probability of event=Number of times event occurred in experimentsTotal number of experiments
the experimental probability is $\frac{3}{20}$320, which is $15%$15%.
How many of the next $50$50 contests can Sweden expect to win?
Just like in the last chapter, we can calculate this by multiplying the experimental probability of an event by the number of trials. In this case Sweden can expect to win
$\frac{3}{20}\times50=\frac{150}{20}$320×50=15020 contests
This rounds to $8$8 contests out of the next $50$50.
To find out how many times an event is expected to happen, we multiply the probability of the event happening with the number of trials:
$\text{Expected number of times an event will occur}=\text{probability of event}\times\text{number of trials}$Expected number of times an event will occur=probability of event×number of trials
Practice questions
Question 4
A retail store served $773$773 customers in October, and there were $44$44 complaints during that month.
Determine, as a percentage, the experimental probability that a customer submits a complaint.
Round your answer to the nearest whole percent.
Question 5
An insurance company found that in the past year, of the $2558$2558 claims made, $1493$1493 of them were from drivers under the age of 25.
Give your answers to the following questions as percentages, rounded to the nearest whole percent.
What is the experimental probability that a claim is filed by someone under the age of 25?
What is the experimental probability that a claim is filed by someone 25 or older?
Question 6
The experimental probability that a commuter uses public transport is $50%$50%.
Out of $500$500 commuters, how many would you expect to use public transport?