# Newsletter

# Another Law of Large Numbers

July 24, 2020

(This was a blog post and not an issue of the newsletter)

I flip a fair coin and it comes up heads. I flip it again and it comes up heads. I ask you to guess the result of the next flip.

What do you guess?

Tails. It's due. It's come up heads twice and it's a fair coin. If it's fair then it should come up tails.

Heads. It's come up heads twice - it's probably not really a fair coin.

Either one. It doesn't matter. If it is a fair coin, there is an equal chance of it coming up heads or tails so it doesn't matter which one we choose.

I'm sorry, I wasn't paying attention. What?

The answer is, of course, that if it's a fair coin (and it is), then no matter what has happened thus far - even if we've had a run of eighteen heads in a row - there is still an equal chance that on this next toss it will come up heads or tails.

We sometimes say that the coin has no memory.

We, on the other hand, see patterns where there aren't any. Sadly, as our current political climate shows, we also don't see patterns where they exist.

In probability theory we would describe the fair coin has a 50% probability of being heads and a 50% probability of being tails. We'd use fractions or decimals and dress this up in some fancy formulas and write

P(Heads) = 0.5

P(Tails) = 0.5

The law of large numbers says that if we ran this experiment a bunch of times (where bunch means lots and lots) and the trials are independent of each other (so the results of one trial doesn't influence another trial) then we expect to get heads roughly half the time and tails roughly half the time.

In other words, say we tossed our coin 100 times. We'd expect to have roughly 50 heads and 50 tails. There are formulas to tell us how closely we expect the number of Heads/total number of tosses to be to 0.5. Those formulas say we expect it to get closer the higher the number of tosses is.

That's essentially the law of large numbers.

So we have two lessons so far: one, knowing the probability of a fair coin never allows us to bet on a single toss with any extra knowledge and two, we feel that's wrong and that a tails is due.

We are not very good at statistics even when we know better. Even with small cases.

But we are good with large numbers. Say we have a fair die. And say that we toss it twelve thousand times. Approximately, how many times should the number six have appeared in those twelve thousand times?

I'll wait while you work this out.

The law of large numbers tells us that if we take the P(Six) which is 1/6 and multiply it by twelve thousand then we get two thousand.

We expect somewhere around two thousand sixes.

We aren't going to be surprised if there are 2100 sixes or 1995 sixes. We'd be pretty surprised if there were only 1000 sixes. We'd be positive, in fact, that there is something wrong with this die - that it isn't fair.

We may be able to more accurately predict outcomes for large numbers than small - but large numbers don't move us.

We can't picture the millions who have had Corona virus or the hundreds of thousands who have died from it.

That's why groups trying to get you to donate to help the starving present you with a picture and a story of one child. We can picture one child. We can't picture thousands.

At this point well over one hundred thousand people in my country have died of the Corona virus in the past six months. More than four million people have had the virus.

And yet many people here dismiss it as fake or overblown because they haven't seen it in their circle of friends. It hasn't impacted them directly.

The coin has come up heads four times in a row - it must be fake.

Corona isn't fake and it isn't fair.

The transmission is not random. We can toss that particular coin in a way that makes it more likely to keep coming up heads.

The trials are not independent. The things we do in this toss impact what happens in the next toss.

We've seen what works in other countries.

We've seen what works in areas of our own country that dramatically reduced the number of cases and deaths.

The only way to beat the house is not to play. Unfortunately, we need everyone not to play to lessen our chances of losing this particular game.

Link to the Podcast episode from August 4, 2023.