In order to make predictions, we sometimes need to determine the probability by running experiments or simulations, or by 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.

Theoretical probability assumes fairness and equal likelihood among possibilities unless stated otherwise. Looking at the experimental probability can help us adjust our predictions and draw conclusions about a situation's outcomes.

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}}

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

Result | Number of rolls | Experimental Probability |
---|---|---|

\ 1 | \ 11 | \dfrac{11}{200}\cdot 100=\ 5.5 \% |

\ 2 | \ 19 | \dfrac{19}{200}\cdot 100=\ 9.5 \% |

\ 3 | \ 18 | \dfrac{18}{200}\cdot 100=\ 9 \% |

\ 4 | \ 18 | \dfrac{18}{200}\cdot 100=\ 9 \% |

\ 5 | \ 20 | \dfrac{20}{200}\cdot 100=\ 10 \% |

\ 6 | \ 114 | \dfrac{114}{200}\cdot 100=\ 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 performances when trying to predict the future. This table shows the winner of the Eurovision Song Contest from 1999 to 2023:

Year | Winning country | Year | Winning country | Year | Winning country |
---|---|---|---|---|---|

\ 1999 | \text{Sweden} | \ 2008 | \text{Russia } | \ 2017 | \text{Portugal} |

\ 2000 | \text{Denmark} | \ 2009 | \text{Norway} | \ 2018 | \text{Israel} |

\ 2001 | \text{Estonia} | \ 2010 | \text{Germany} | \ 2019 | \text{Netherlands} |

\ 2002 | \text{Latvia} | \ 2011 | \text{Azerbaijan} | \ 2020 | \text{Contest cancelled} |

\ 2003 | \text{Turkey} | \ 2012 | \text{Sweden} | \ 2021 | \text{Italy} |

\ 2004 | \text{Ukraine} | \ 2013 | \text{Denmark} | \ 2022 | \text{Ukraine} |

\ 2005 | \text{Greece} | \ 2014 | \text{Austria} | \ 2023 | \text{Sweden} |

\ 2006 | \text{Finland} | \ 2015 | \text{Sweden} | ||

\ 2007 | \text{Serbia} | \ 2016 | \text{Ukraine} |

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 24 in total. The winning country is the event, and we can tell that 4 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{4}{24}, which is about 17\%.

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 |

a

How many meals were ordered altogether?

Worked Solution

b

Determine the experimental probability, as a ratio, that a customer will order a beef meal.

Worked Solution

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

b

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

Worked Solution

Idea summary

\text{Experimental probability of event} = \dfrac{\text{Number of times event occurred}}{\text{Total number of experiments}}

Now, we will use the applet shown to compare the experimental and theoretical probabilities of an experiment.

Click the "Roll the die 6 times" button. How do the experimental probabilities compare to the theoretical probabilities?

Reset the frequencies, then click the "Roll the die 60 times" button. How do the experimental probabilities compare to the theoretical probabilities? Are the results closer than when you rolled it 6 times?

Reset the frequencies, then click the "Roll the die 600 times" button. How do the experimental probabilities compare to the theoretical probabilities now?

What do you notice about the probabilities as the number of trials increases?

The experimental probability does not always equal the theoretical probability.

As we compare theoretical and experimental probabilities and conduct more trials, we begin to notice a pattern. The more trials that we run, the closer the experimental probabilities will be to the theoretical probabilities of the event. This is known as the **law of large numbers**.

Suppose you want to know if flipping the coin shown truly results in an expected probability of 50\% for heads and 50\% for tails.

The table shows the number of flips and the actual frequency of heads and tails.

Number of Flips | Heads | % | Tails | % |
---|---|---|---|---|

10 | 7 | 70\% | 3 | 30\% |

20 | 13 | 65\% | 7 | 35\% |

50 | 29 | 58\% | 21 | 42\% |

100 | 54 | 54\% | 46 | 46\% |

1000 | 490 | 49\% | 510 | 51\% |

As the number of flips increase, we see that the percentages are becoming closer and closer to the theoretical probability or 50\% for each side of the coin.

A trial is to be conducted by flipping a coin.

a

What is the theoretical probability of flipping tails on a coin?

Worked Solution

b

A coin was flipped 184 times with 93 tails recorded.

What is the exact experimental probability of flipping tails with this coin?

Worked Solution

c

Compare the theoretical and experimental probabilities of flipping tails.

Worked Solution

Anasofia and Caio each surveyed students from their school to see how many people would be interested in a fun run event. The results of their surveys are shown in the table.

Anasofia's survey | Caio's survey | |
---|---|---|

Would join the fun run | 54 | 408 |

Would not join the fun run | 89 | 460 |

Anasofia claims that about 38\% of the students would join the fun run based on her survey results. Caio claims that about 47\% of the students would join the fun run based on his survey results.

Which claim do you think is more valid? Explain your reasoning.

Worked Solution

Idea summary

The law of large numbers states that as the number of trials increase, the experimental probability gets closer to the theoretical probability.