Here is my cat, Albert.
Albert is a cat with many friends.
His 3 favorite ones are: Oscar, Max and Émilie.
The other day, Oscar suggested a game.
A coin has to be tossed till it lands on head, with a maximum of 3 tosses.
In case of a favorable outcome, Oscar promises to give some of his chips.
Otherwise, the player having tossed the coin must give some chips to Oscar.
The rules of the game are as follow.
If the cat tossing the coin ends up with a head after the first toss,
the game stops and he receives two of Oscar’s chips.
Should he end up with a tail and then a head, Oscar will give him one chip.
But in case of two tails and then a head, it is he who must give one chip to Oscar.
Even worse, if the player ends up with head three times in a row,
then he must give no less than 10 chips to Oscar.
Albert does not want to lose his chips.
He wonders what are the odds to fall in one of the last two cases.
The probability of a coin to land on head being 1/2,
he calculates with Max and Émilie that the probability of three heads in a row is the cube of 1/2, thus 1/8.
Using the same reasoning, Albert concludes that the probability of two heads and then a tail is 1/8, too
Overall, the probability of losing Oscar’s game is therefore 1/8 plus 1/8, that is, 1/4 only.
Phew! Albert feels better.
He decides to start playing.
He ends up with a head and a tail.
Oscar, as promised, gives him one of his chips.
Meanwhile, Max ends up with a head after one toss.
He therefore receives one chip from the stack of Oscar.
Finally, it is Émilie’s turn.
She gets the same outcome as Albert and therefore receives one chip.
Albert, Max and Émilie now feel sorry for Oscar. As expected, he lost.
Yet, to their surprise, Oscar offers a second round. Has Oscar lost his mind?
Do you wish to make money through games of chance?
Don’t go to the casino, just open one!
But what is the secret of this kind of business?
Albert, Max and Émilie are convinced that only the balance between probability of winning
and probability of losing matters.
But Oscar is ahead of them.
He knows that things are not that simple.
What is the proper way to approach this problem then?
How can one tell if a game of chance is interesting for the player?
Before we go further, we need to introduce an important concept of statistics:
Expected value, sometimes called average, is calculated for a random variable.
Do you remember the random variable you found in the attic, Albert?
I’ve got the blueprint back!
The domain of this random variable was actually made of four elements: -4, 0, 1 and 2.
The related weights were 0.005, 0.535, 0.14 and 0.32
Let us calculate the expected value.
We need to multiply each value of the domain by its weight and to sum everything,
getting this way the value of 0.76
So what? Patience Albert, we’re getting there.
Let’s recall how a random variable works.
A random variable is a bit like a box: every time you open it, a number comes out.
Statisticians then say that one observation of the variable has been made.
Problem: it is impossible to say in advance what value will actually come out.
We know it’s going to be a value from the domain, but we never know which one.
For long, statisticians thought that there was nothing to add to the subject.
And then, Jacob Bernoulli made and extraordinary discovery.
Even though no single observation from a random variable can be predicted,
the sum of a large number of observations can,
and the accuracy of this prediction increases with the number of observations made.
You want an example, Albert?
Here are the 100 observations of the variable from the attic that you made a few months ago.
Even though these observations are each random,
there is something one can predict: their sum should be close to 76.
And indeed, if we try, the sum is… 80!
We miss the predicted target of only 4 units.
Why 76? Because the expected value of the random variable is 0.76
This value of 0.76 is the value that, conceptually, comes out of the box each time someone opens it.
Opening it a hundred times means ending up with the value 0.76 a hundred times, thus with 76.
Let’s push this reasoning further: what can we say when the box is opened a thousand times?
The sum of a thousand observations should be close to 760.
Even better, because we now work with a thousand observations,
we can say that the sum of these 1000 observations should be closer to 760
than 80 was to 76 in the previous case.
Jacob Bernoulli thus discovered that,
even though successive observations from a random variable are unpredictable,
the sum of a large number of these observations is quite predictable,
and is related to the random variable’s expected value.
Even today, this is considered to be one of the greatest statistical discovery ever made.
It is however Mr. Poisson who gave the name in use today for this phenomenon:
the law of large numbers (LLN).
Thanks to the law of large numbers, we can hack the secret behind any game of chance.
Let’s try with Oscar’s game. What is the expected value?
The game of Oscar can be viewed as a random variable with a domain of 2, 1, -1 and -10, in chips.
The related weights are 1/2, 1/4, 1/8 and 1/8.
We calculate the expected value:
each value of the domain is multiplied by its weight before the sum of everything is computed.
The result is then -0.125 chip, that is, minus one eighth.
The expected value of Oscar’s game is lower than 0!
Things are quite clear: each time a cat plays at Oscar’s game,
it is as if one eighth of a chip is lost by that cat.
If Oscar manages to make his friends play a large number of times,
the law of large numbers ensures him to win.
After 80 tosses, he should have 10 extra chips.
And after 800 tosses, this becomes 100 extra chips!
Mmmmh. Albert has put a lot of thought into all this.
Unsurprisingly, he does not want a second turn to Oscar’s game.
He has something else in mind.
A new game of chance, this time based on the roll of dice.
The player rolls two dice and if the sum of the two values is an even number,
then this is the number of extra chips the player gets.
For instance, if two and four are the numbers, the player gets 6 chips.
If the numbers are 3 and 5, the player gets 8 chips.
In the best case, the player gets 12 chips, when the rolled dice each show six pips.
However, in case the sum of the two values is an odd number, the player loses 7 of his chips.
These are Albert’s rules.
Dear viewer, what do you think about Albert’s game?
Would you play it?
Leave us your answer in the comment section of this video, on Twitter or on Facebook!
Do you wish to see Albert in other adventures?
Check out his other videos and do not forget to subscribe to his Youtube channel!