12.06 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.

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:

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:

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:

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.

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:

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. 

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.

Practice questions

Question 1

Question 2

Question 3