Define the Law of Large Numbers, citing a credible source.The Law of Large Numbers says that as more and more trials of an experiment are performed, the percentage error between the theoretical probability and the experimental results will decrease towards zero.
B. Explain the Law of Large Numbers in your own words using a coin toss as an example.
Theoretically, a coin lands heads up 50% of the time and tails up 50% of the time. If we flip the coin enough times, the Law of Large Numbers says that our heads and tails counts will get closer and closer to being equal to the 50/50, which is our expected value
C. Apply the Law of Large Numbers
1. Using a coin toss and fictitious data, explain how the following scenario is possible: As the number of trials increases, the differences between the number of actual and expected successes tends to grow, but the difference between the percentage of actual and expected successes tends to decrease.
We flip a coin 10 times. We get 7 heads and 3 tails. We would expect, from the theory, to get 5 of each, so our numerical difference is 2. The percentage difference is 2/5 = 40%. Therefore, on a small number of flips, we had a small numerical difference (2), but a large percentage difference (40%).
Now, say you flip the coin 1000 times. In this experiment, you get 481 heads and 519 tails. We would have expected to get 500 of each, so our numerical error is 19. The percentage error is now 19/500 = 3.8%. Therefore, in our experiment with a large number of flips, the numerical error is larger (it went from 2 to 19), but the percentage error has decreased dramatically, from 40% down to 3.8%.
The Law of Large Numbers only relates to the percentage error decreasing with more flips performed. This little experiment has supported our belief in the Law. 2. Explain your answer to the following question: Is it true that if I flip a coin 1,000 times I will get heads 500 times? If a coin is flipped 1000 times, it will be quite extraordinary to get precisely 500 heads and 500 tails. The Law of Large Numbers only says that the percentage error will get less and less as you flip more times, and as I discussed in the previous part, the numerical error actually tends to grow over time.
3. Explain your answer to the following question: Is it true that if I get tails 3 times in a row that my chances of getting heads on my next toss is greater than 50%?
It is not true that if you flip a coin and you get 3 tails in a row, your next flip is more likely to be a heads. The coin itself has no memory of past events, so it cannot know you had three tails in a row. Each flip is independent of each trial.
Grimmett, G. R. and Stirzaker, D. R. (1992). Probability and Random Processes, 2nd Edition. Clarendon Press, Oxford. ISBN 0-19-853665-8