10.01 Experimental probability

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. It can also be called relative frequency. 

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. 

Worked examples

example 1

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

(a) 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%.

(b) How many of the next $50$50 contests can Sweden expect to win?

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.

Example 2

$500$500 cables were tested at a factory, and $76$76 were found to be faulty.

(a) What is the experimental probability that a cable at this factory will be faulty?

$P\left(\text{Faulty}\right)$P(Faulty)	$=$=	$\frac{\text{Number of favourable outcomes}}{\text{Total number of trials}}$Number of favourable outcomesTotal number of trials​
	$=$=	$\frac{76}{500}$76500​
	$=$=	$0.152$0.152

(b) If $1500$1500 more cables were tested, how many would you expect to be faulty?

Now that we know that approximately $15.2%$15.2% are faulty (from our experimental data), we could expect the same percentage to be faulty from any amount.

So:

$15.2%$15.2% of $1500$1500$=$=$0.152\times1500=228$0.152×1500=228

We could expect $228$228 to be faulty from $1500$1500 cables.  

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5