Law of Large Numbers
Experimental Probability
the probability that is determined on the basis of the results of an experiment repeated many
times
Theoretical Probability
the probability that is determined on the basis of reasoning
Theoretical probability and experimental probability are two methods for
calculating the probability of an event occurring. Theoretical probability is
determined by mathematical principles, whereas experimental probability is
determined by direct observation or experimentation.
Theoretical probability and experimental probability are related in that
theoretical probability provides a baseline or ideal expectation, whereas
experimental probability provides a more realistic assessment of what actually
happens in practice.
SIMULATION
1. The theoretical probability of the free throw shooter is 80%. Meaning there should be an 80%
chance that the shooter's free throw would be successful. However, using the spreadsheet.
Performing for 100 trials. (twelve 9’s, eleven 10’s, totaling 23 failures, or 23% failures). Meaning
the experimental probability was 77%.
2. The first two failures in a row were attempts 12 and 13. the first and only three failures in a row
were attempts 71 - 73 all being 9’s.
Law of Large Numbers
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GAMES FAIR
Game
Theoretical
Probability (as a
percent)
Experimental
Probability of 10 trials
(as a percent)
Experimental
Probability of 50
trials(as a percent)
Experimental
Probability of 100
trials(as a percent)
Spinner: Win 10
or more points.
4/10 = 40%
2/10 = 20%
17/50 = 34%
38/100 = 38%
Single Card Flip:
Win 15 points.
18/52 = 34.6%
3/10 = 30%
12/50 = 24%
31/100 = 31%
Application and Communication - Explaining The Law of Large Numbers
Law of Large Numbers
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The Law of Large Numbers is a mathematical rule stating that as the sample size of a random
event grows, the average of the observed results approaches the true expected value.
A casino benefits from the Law of Large Numbers by providing games with a house edge, which
means that the expected value of each bet is in the casino's favour. While individual players may
win or lose in the short term, due to the law of large numbers, the casino will make a profit over a
large number of bets.
TWO DICE
EVENT
Predicted
Theoretical
Probability (as a
percent)
Experimental
Probability after
100 trials (as a
percent)
Experimental
Probability after
200 trials (as a
percent)
Experimental
Probability after
300 trials (as a
percent)
Experimental
Probability after
500 trials (as a
percent)
SUM OF 6
5/36 = 13.8%
13/100 = 13%
32/200 = 16%
44/300 = 14.7%
71/500 = 14.2%
SUM OF 7
6/36 = 16.7%
21/100 = 21%
28/200 = 14%
48/300 = 16%
75/500 = 15%
SUM OF 8
5/36 = 13.8%
13/100 = 13%
27/200 = 13.5%
41/300 = 13.7%
69/500 = 13.8%
Law of Large Numbers
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