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17.05 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, since we determine the probability of each outcome by looking at past events.

Experimental probability

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):

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

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

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

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?

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

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

This rounds to 8 contests out of the next 50 .

Examples

Example 1

An insurance company found that in the past year, of the 2558 claims made, 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.

a

What is the experimental probability that a claim is filed by someone under the age of 25?

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}\displaystyle =\displaystyle \frac{1493}{2558}Substitute the given values
\displaystyle =\displaystyle \dfrac{1493}{2558} \times 100\%Multiply by 100\%
\displaystyle =\displaystyle 58 \%Round to the nearest whole percent
b

What is the experimental probability that a claim is filed by someone 25 or older?

Worked Solution
Create a strategy

Find the numbers of the claims were made by people who are 25 or older and use the formula for experimental probability.

Apply the idea

The number of claims made by drivers 25 or older is 2258- 1498 = 1065.

\displaystyle \text{Experimental probability}\displaystyle =\displaystyle \dfrac{1065}{2558}Substitute the values
\displaystyle =\displaystyle \dfrac{1065}{2558} \times 100\%Multiply by 100\%
\displaystyle =\displaystyle 42 \%Round to the nearest whole percent

Example 2

The experimental probability that a commuter uses public transport is 50\%. Out of 500 commuters, how many would you expect to use public transport?

Worked Solution
Create a strategy

To find the expected number of commuters multiply the total number of commuters by the probability.

Apply the idea
\displaystyle \text{Expected commuters}\displaystyle =\displaystyle 500 \times 50\%Multiply by a probability
\displaystyle =\displaystyle 500 \times \dfrac{50}{100}Convert to a fraction
\displaystyle =\displaystyle 250Evaluate
Idea summary

To find the experimental probability of an event we can use the formula:\text{Experimental probability} = \dfrac{\text{Number of times event occurred in experiments}}{\text{Total number of experiments}}

Outcomes

MA4-21SP

represents probabilities of simple and compound events

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