Hi this is Presh Talwalkar.
Warren Buffett once challenged Bill Gates
to a special dice game like the following:
the dices each had six faces that were equally likely to show up
but the numbers were different than a standard dice.
One of the dices had all 3s on all of its six faces.
Another one had 0 on two of its faces and 4 on four of its faces.
A third dice had an equal number of 1s and 5s,
and a fourth dice had 2 on four of its faces and 6 on two of its faces.
So here's how the game works:
you and I are each going to pick a different one of these dices.
We're not going to pick the same dices; we're going to pick different dices.
We're then going to roll against each other,
and the person who gets the higher number gets the point.
You then repeat that many many times so that the person who rolls the highest number of points over many rolls is going to win the game.
I'm going to graciously offer you to let you pick your dice first.
Which one do you want?
Give this problem a try and when you ready keep watching the video for the solution
So let's analyze this game, suppose you pick the dice with all threes.
What I am going to do is I am going to pick the dice which has zero on two of its spaces, and four on four of its spaces.
What's going to happen if we roll against each other?
Well, if I roll a zero which happens two of the sixth times, I'm going to lose to you because you are always going to roll a three.
However, in the four out of the six times I roll a four I'm actually going to beat you.
So I'm going to beat you, two-thirds of the time, so if you pick a dice which has all threes, you're going to lose to me two-thirds of the time.
So now suppose you know this strategy and you say, "Well, why don't I pick the dice that has two zeros and four fours?"
Well if you are going to pick that dice, then I am going to pick the dice that has three ones and three fives.
It's a little harder to analyze what happens when we roll these two dice
against each other but it's not too complicated.
Now, I'm going to win if I roll a five because anytime I roll a five it's higher than you rolling a zero or a four.
So in three out of the six cases I'm going to roll five and I'm guaranteed to win.
Now when I roll the one, in three out of six cases,
I'm only going to beat you in the two out of the six cases that you roll a zero as well.
So in summary,
I'm going to win 3/6 times 1 when I roll a 5,
plus 3/6, when I roll a zero--when I roll a ONE, times 2/6 when you roll a zero.
So this nets out to 2/3
of the time, I'm going to win. So
once again, your die that you pick
is going to lose to my die 2/3 of the time.
This argument continues. Let's suppose you pick
the die that I just had,
Well in that case I'm going to pick the next die which has
2 on four of its faces and 6 on two of its faces.
We can do a similar calculation.
I'm going to win two out of the six times when I roll a 6,
guaranteed.
And I'm also going to win if I roll a 2,
which is 4/6 of the time,
against your 1, which happens 3/6 of the time.
So once again, I'm going to beat your die 2/3 of the time.
Now I'm going to win whenever I roll--
--whenever YOU roll a 2, which happens 4/6 of the time,
so once again your die is losing to my die 2/3 of the time.
So what we've shown
is that each die
is actually going to lose 2/3 of the tme to another die.
And this is what's known as non-transitive dice.
Each die can be beaten like another
like a game of rock, paper, scissors.
This is a cyclical game, and that's why it's non-transitive.
The first person who selects the die is always going to be at a
disadvantage to someone who picks
this second
because each can be beaten by another.
As the story goes, Warren Buffet challenged Bill Gates to this and said,
"Go ahead. Why don't you select first?"
Bill Gates was smart enough to say, "Let me inspect the dice."
And he figured out that there was something weird going on with the dice.
Thanks for watching.
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