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Stage 5.1

8.01 Experimental probability

Lesson

Introduction

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 (or relative frequency), since we determine the probability of each outcome by looking at past events.

Experimental probability

A six sided die which is loaded. Ask your teacher for more information.

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).

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 \dfrac{1}{6}.

Instead we will need to roll the die many times and record our results, and use these results to predict the future.

ResultNumber of rolls
\ 1 \ 11
\ 2 \ 19
\ 3 \ 18
\ 4 \ 18
\ 5 \ 20
\ 6 \ 114

Here are the results of an experiment where the die was rolled 200 times.

We can now try to predict the future using this experimental data, and the following formula:\text{Experimental probability of event} = \dfrac{\text{Number of times event occurred in experiments}}{\text{Total number of experiments}}

Here is the table again, with the experimental probability of each face listed as a percentage:

ResultNumber of rollsExperimental Probability
\ 1 \ 11 \ 5.5 \%
\ 2 \ 19 \ 9.5 \%
\ 3 \ 18 \ 9 \%
\ 4 \ 18 \ 9 \%
\ 5 \ 20 \ 10 \%
\ 6 \ 114\ 57 \%

A normal die has around a 17\% chance of rolling a 6, but this die rolls a 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:

YearWinning countryYearWinning country
\ 1999 \text{Sweden}\ 2009 \text{Norway}
\ 2000 \text{Denmark} \ 2010\text{Germany}
\ 2001 \text{Estonia} \ 2011\text{Azerbaijan}
\ 2002 \text{Latvia}\ 2012\text{Sweden}
\ 2003 \text{Turkey}\ 2013\text{Denmark}
\ 2004 \text{Ukraine} \ 2014\text{Austria}
\ 2005 \text{Greece} \ 2015\text{Sweden}
\ 2006 \text{Finland}\ 2016\text{Ukraine}
\ 2007 \text{Serbia}\ 2017\text{Portugal}
\ 2008 \text{Russia }\ 2018\text{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 in total. The winning country is the event, and we can tell that 3 of the contests were won by Sweden. So using the same formula as above, \text{Experimental probability of event} = \dfrac{\text{Number of times event occurred in experiments}}{\text{Total number of experiments}}

the experimental probability is \dfrac{3}{20}, which is 15\%.

How many of the next 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

\displaystyle \dfrac{3}{50} \times 50\displaystyle =\displaystyle \dfrac{150}{20} \text{ contests}

This rounds to 8 contests out of the next 50 .

The theoretical probability of rolling any given number on a fair die is \dfrac{1}{6}. However, if we rolled the die 100 times and got these results:

ResultNumber of rollsExperimental Probability
\ 1 \ 14 \ 14 \%
\ 2 \ 19 \ 19 \%
\ 3 \ 18 \ 18\%
\ 4 \ 20 \ 20\%
\ 5 \ 16 \ 16 \%
\ 6 \ 13\ 13\%

We can see that the experimental values even for a fair die might not equal the theoretical probability of 16.67\%. If we convert this to the percentage of the rolls though we can see they are reasonably close to the theoretical probability. But perhaps the die could be a little bit biased. In this case we can roll the die more to see if the experimental probability approaches the the theoretical probability with a higher number of rolls.

ResultNumber of rollsExperimental Probability
\ 1 \ 171 \ 17.1 \%
\ 2 \ 159 \ 15.9 \%
\ 3 \ 172 \ 17.2 \%
\ 4 \ 166 \ 16.6 \%
\ 5 \ 169 \ 16.9 \%
\ 6 \ 163\ 16.3 \%

If we rolled the die 1000 times we should expect the percentage to get closer, but not the actual number of rolls will still not be equal to the expected number of rolls.

Examples

Example 1

Ryan decided to flip a coin 20 times.

a

How many times would he expect a head to appear?

Worked Solution
Create a strategy

Find the probability then multiply it by the number of trials.

Apply the idea

The theoretical probability of flipping a head is \dfrac{1}{2}.

\displaystyle \text{Number of times}\displaystyle =\displaystyle \frac {1}{2} \times 20Multiply by the number of trials
\displaystyle =\displaystyle 10Evaluate

Ryan can expect that a head will appear 10 times.

b

After he finished flipping the coins, he noticed that heads had appeared 11 times. Write the experimental probability of getting a head as a fraction.

Worked Solution
Create a strategy

Use the formula for experimental probability.

Apply the idea
\displaystyle \text{Experimental probability}\displaystyle =\displaystyle \dfrac{\text{Number of times event occurred}}{\text{Number of experiments}}Write the formula
\displaystyle =\displaystyle \dfrac{11}{20}Substitute the values

Example 2

To prepare for the week ahead, a restaurant keeps a record of the number of each main meal ordered throughout the previous week.

MealFrequency
\text{Chicken}21
\text{Beef}55
\text{Lamb}44
\text{Vegetarian}31
a

How many meals were ordered altogether?

Worked Solution
Create a strategy

Add the frequencies in the table together.

Apply the idea
\displaystyle \text{Total meals}\displaystyle =\displaystyle 21+55+44+31Add the frequencies
\displaystyle =\displaystyle 151Evaluate

There were 151 meals ordered altogether.

b

What is the experimental probability that a customer will order a chicken meal? Round your answer to the nearest whole percent.

Worked Solution
Create a strategy

Use the formula for experimental probability and convert to a percentage by multiplying by 100\%.

Apply the idea
\displaystyle \text{Experimental probability of event}\displaystyle =\displaystyle \frac{21}{151}Substitute the given values
\displaystyle =\displaystyle \dfrac{21}{151} \times 100\%Multiply by 100\%
\displaystyle =\displaystyle 14 \%Evaluate and round
Idea summary

To calculate the experimental probability:\text{Experimental probability of event} = \dfrac{\text{Number of times event occurred}}{\text{Total number of experiments}}

To calculate the expected number of trials with a specific outcome:

\text{Expected no. of favourable trials} = \text{Experimental probability} \times \text{Total no. of trials}

Experimental probability does not need to equal theoretical probability. Experimental probability should get closer to the theoretical probability value as the number of trial increases.

Outcomes

MA5.1-13SP

calculates relative frequencies to estimate probabilities of simple and compound events

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