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Middle Years

12.06 Experimental probability

Lesson

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.

Experimental probability

 To calculate the experimental probability:

$\text{Experimental probability of event}=\frac{\text{Number of times event occurred in experiments}}{\text{Total number of experiments}}$Experimental probability of event=Number of times event occurred in experimentsTotal number of experiments

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

$\text{Expected number of favourable trials}=\text{Experimental probability of event}\text{Total number of trials}$Expected number of favourable trials=Experimental probability of eventTotal number of trials

 

Number of trials

The theoretical probability of rolling any given number on a fair die is $\frac{1}{6}$16. However, if we rolled the die $100$100 times and got these results:

Result Number of rolls Experimental probability
$1$1 $14$14 $14%$14%
$2$2 $19$19 $19%$19%
$3$3 $18$18 $18%$18%
$4$4 $20$20 $20%$20%
$5$5 $16$16 $16%$16%
$6$6 $13$13 $13%$13%

We can see that the experimental values even for a fair die might not equal the theoretical probability of $16.67%$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. 

Result Number of rolls Experimental probability
$1$1 $171$171 $17.1%$17.1%
$2$2 $159$159 $15.9%$15.9%
$3$3 $172$172 $17.2%$17.2%
$4$4 $166$166 $16.6%$16.6%
$5$5 $169$169 $16.9%$16.9%
$6$6 $163$163 $16.3%$16.3%

If we rolled the die $1000$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.

Experimental probability and theoretical probability

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.

 

Practice questions

Question 1

Ryan decided to flip a coin $20$20 times.

  1. How many times would he expect a head to appear?

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

Question 2

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

Meal Frequency
Chicken $21$21
Beef $55$55
Lamb $44$44
Vegetarian $31$31
  1. How many meals were ordered altogether?

  2. What is the experimental probability that a customer will order a chicken meal?

    Give your answer as a percentage, rounded to the nearest whole percent.

Question 3

A mixed martial arts club posted the results from the tournaments last year. The table shows the main fighting style of the winner for each tournament.

Event Frequency
Karate $40$40
Wrestling $55$55
Judo $46$46
Jiu-Jitsu $59$59
Event Frequency
Karate $40$40
Wrestling $55$55
Judo $46$46
Jiu-Jitsu $59$59
  1. What were the total number of tournaments?

  2. What is the experimental probability of a wrestler winning?

  3. Which is the most successful fighting style?

    Judo

    A

    Jiu-Jitsu

    B

    Wrestling

    C

    Karate

    D

    Judo

    A

    Jiu-Jitsu

    B

    Wrestling

    C

    Karate

    D
  4. If $1000$1000 further tournaments are to be held, how many tournaments would you expect Judo fighters to win?

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