8.03 Experimental probability
Introduction
In the previous lesson we looked at theoretical probability , which is ratio of the number of favorable outcomes to the total number of possible outcomes. In this lesson we are going to explore experimental probability, which is based on events that have occured.
Experimental probability
In order to make predictions, 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. You may also see the term relative frequency which is the same as the 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):
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 future outcomes. Here are the results of an experiment where the die was rolled 200 times:
| Result | Number of rolls |
|---|---|
| \ 1 | \ 11 |
| \ 2 | \ 19 |
| \ 3 | \ 18 |
| \ 4 | \ 18 |
| \ 5 | \ 20 |
| \ 6 | \ 114 |
We can now try to make predictions using this experimental data, and the following formula:\text{Experimental Probability} = \dfrac{\text{Number of times event occurred}}{\text{Total number of experiments}}
Let's find the experimental probability for rolling a 1 as a percentage.
| 1 | \displaystyle \text{Experimental Probability} | \displaystyle = | \displaystyle \dfrac{11}{200} | subsitute the number of times that 1 was rolled and the total number of rolls into the formula |
| 2 | \displaystyle \text{Experimental Probability} | \displaystyle = | \displaystyle 0.055 | divide the numerator by the denominator |
| 3 | \displaystyle \text{Experimental Probability} | \displaystyle = | \displaystyle 5.5\% | convert to a percentage by multiplying by 100\% |
Here is the table again, with the experimental probability of each face listed as a percentage:
| Result | Number of rolls | Experimental Probability |
|---|---|---|
| \ 1 | \ 11 | \ 5.5 \% |
| \ 2 | \ 19 | \ 9.5 \% |
| \ 3 | \ 18 | \ 9 \% |
| \ 4 | \ 18 | \ 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 2009 to 2018:
| Year | Winning country | Year | Winning 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}}{\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 previous lesson, 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} |
| \displaystyle \dfrac{150}{20} | \displaystyle = | \displaystyle 7.5 \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.
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?
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:
| Meal | Frequency |
|---|---|
| \text{Chicken} | 54 |
| \text{Beef} | 32 |
| \text{Lamb} | 26 |
| \text{Vegetarian} | 45 |
How many meals were ordered altogether?
Determine the experimental probability, as a percentage, that a customer will order a beef meal.
Round your answer to the nearest whole percent.
\text{Experimental probability of event} = \dfrac{\text{Number of times event occurred}}{\text{Total number of experiments}}You may also see the term relative frequency which is the same as the experimental probability.