Question

Use an example object like a dice, coin, spinner, etc. to thoroughly explain what the Law of Large Numbers tells us as it relates to theoretical probabilities.

Answer 1

solution:

The law of large numbers, in probability and statistics, states that as a sample size grows, its mean gets closer to the average of the whole population. 0r

The law of large numbers in probability theory. It states that, as a probabilistic process is repeated a large number of times, the relative frequencies of its possible outcomes will get closer and closer to their respective probabilities.

For example, flipping a regular coin many times results in approximately 50% heads and 50% tails frequency, since the probabilities of those outcomes are both 0.5.

Let’s say you flip a regular coin a certain number of times. After each flip you record if it was heads or tails. At the end you calculate the total count of the 2 outcomes. Here’s a question . About what % of the the flips do we think will be heads?

our feeling is to say “about 50%”, yes? That’s because we know regular coins have about 0.5 probability of landing heads. our reasoning is correct, but still, let’s try to apply some skepticism.

I said you flipped the coin a certain number of times but I didn’t say how many. What if you only flip it once? Then, if it lands heads, 100% of the flips will be heads, otherwise 0% will be heads. Those are clearly not the expected 50%!

What if you flip it, say, 4 times? Well, grab a coin and try this experiment on your own. You’ll see that it’s not that uncommon to get sequences with 3 or even 4 heads/tails. In such cases, you will still get relative frequencies like 0%, 25%, and 75%. As you can see, the “about 50%” answer is at least a little suspicious.

Of course, when I said that you flip the coin “a certain number of times”, you probably assumed I meant a higher number, like 100 or 1000. Then, you say, the relative frequency of heads flips will be much closer to 50%. And I agree.

In fact, this is exactly what the law of large numbers is all about.

When you flip the coin 1, 2, 4, 10, etc. times, the relative frequency of heads can easily happen to be away from the expected 50%. That’s because 1, 2, 4, 10… are all small numbers. On the other hand, if you flip the coin 1000 or 10000000 times, then the relative frequency will be very close to 50%, since 1000 and 10000000 are large numbers.