We're gonna go gambling. We're creating a-
let's call it a fictional casino
and we're playing a game with a 50/50
chance of winning.
Okay, so we're doing roulette - red or black so
there's no zero, so it is
actually a 50/50 chance red or black.
What we're going to look at is something
called the Martingale Strategy.
Every time you lose a bet you double
down
on the next bet; and therefore eventually you will
win - is the idea. - (Brad: What do you mean by)
(double down?)
So if you bet one pound and lose then
you bet again
and you would bet two pounds; and then if
you lost again then you go four pounds;
if you lose you go eight pounds. So you
double the last bet you made.
Provided you have pretty much unlimited
money you will eventually
win. A win is: come out of our fictional
casino with more money than I went in.
(You want to make a profit.) - Exactly we're
making a profit. - (Should we say we're)
(always betting on red?)
(Or black or-?) - I think- I'm
tempted to go black.
So I write with a black pen. I think if I
was given the choice I would- I would I'd
like a-
I'm going for black over red I think.
(Okay so we're betting on black.) - Betting
on black. And we're going to keep going;
you think about well when is the first
time I lose? So if I lose at step k
I've had my bet at step 1,
and I bet one pound, and then I must have lost
because we're assuming we're going to
keep going until this number k. Then at
step 2 I bet two pounds. Then step three -
again we're assuming I've lost at step
2 - I double my bet
so this is now going to be four pounds.
Step 4 is going to be double that, eight
pounds, and this
carries on. - (If you win your money gets)
(doubled?) - If I win my money will be
doubled, yes, it's a 50/50 game
such that whatever I bet- like in, exactly
like roulette. We don't have a zero
but the rules otherwise are the same. If
I bet on black, which is what we're doing,
if it lands on black I double my money.
(Because that zero is how casino is insured.)
The zero- exactly. You know this way. The zero
is how- casinos know this and this is why
they have zero, because it's never a
50/50 chance.
But our mathematically fair casino gives
us a true 50/50 chance. At step
k then we can figure out how much we would be
betting because there's a pattern here.
So at step 1
I bet one pounds, step 2 two pounds,
step 3 four pounds.
So what's happening with our bets is
these are increasing powers of 2.
So this is times 2, times 2, times 2.
So the actual formula
is going to be 2 to the power of k
minus 1. So we can just check this: so
at step 1, so k is 1; 2 to the power
of 0 is 1.
Step 2 k is 2, 2 to the power 1
is 2;
2 squared is 4, 2 cubed is 8 so at
step k we have just bet
2 to the k minus 1 pounds.
If I then lose at step k my total loss
so far is the sum of all of these bets;
so 1 plus 2
plus 4 plus 8 all the way up to 2
to the k minus 1. We can write this as
the sum from
j equals 0 to k minus 1 of 2
power j. So 2 to the 0 is 1 then
plus
2 to the 1 is 2 plus 2 squared is
4. So this is a shorthand way of writing
out this sum.
And this is what we call a geometric
series; we
have a formula for the sum of a
geometric series so
we can plug it in. It's the first term, which is
1, multiplied by 1 minus
the next term in the sequence. So it ends
at 2 to the k minus 1
so if we were to go to the next step
that would be 2 to the k;
and then we divide by 1 minus
the common ratio between the terms.
So what we're multiplying by at each
step, so divided by 1 minus 2. So you
tidy this up
and this is 2 to the k minus
1. So this is the total amount of money
I have lost when I lose my bet at step
k. So in the next step I'm going to bet
2 to the k pounds - I'm going to win
hopefully
and then that covers my losses with one
pound to spare.
So after all of this effort, after losing
at some step
if I follow the Martingale strategy I
will eventually win -
we hope - before I run out of money, that's
a whole other question,
and I will win one pound. So we're not
exactly-
like it's good, it's a profit, but it's
it's not like- we want more than a pound
don't we really? Like you don't really go
to a casino to make a pound.
(But if you scaled it to 10 pounds or a)
(million pounds as)
(as your- like your multiple)
(you you're guaranteed to win a million pounds!)
Yeah, so so to win a million pounds
you've got to
go through this strategy one million
times, winning one pound each time.
(Or just make that a million pound bet;)
(a two million pound bet; a four million)
(pound bet, all the way to a two)
(to the minus 1 times a million-) - We
could also do that yeah. So we could also change
the amount we bet at the beginning and
then that would be-
the amount that you your initial stake
is the amount that you'll win
through this strategy. So this strategy
is going to win you,
through this process, we're going to win
a pound each time. Now,
there are problems with this. There is a
reason that this is not a foolproof
gambling strategy. First of all
most games aren't 50/50; if we're doing
roulette there's a zero which is green,
so it's not red or black, so you've not
actually got a 50/50 chance.
And that kind of really alters what's
what's happening. - (In some casinos there)
(are two zeros)
Yes zero and double zero, exactly yeah, so
they're even- they're even
stingier those odds. The other thing is
casinos have maximum bet limits.
In most casinos you can't walk in and
just drop a million pounds,
like they're gonna be like no there's a
maximum bet limit here.
And even if you start really low and you
keep losing
this will grow really quickly. This is
exponential growth which
we know a lot about these days. And this
will grow really quickly
and it will actually reach a million a
lot faster than you realise. - (So if red comes up)
(lots of times in a row - ?) - Yeah you're going
to reach the maximum betting limit in an
actual casino and then
you're screwed, you've then lost like a
million pounds, you've not got a chance
to get your money back.
Because that's the key; if this fails
you've lost
2 to the power k minus 1 pounds which is
a lot of money if
k is like 10 or something, like, so it can
really really ramp up.
(Just for the sake of clarity too Tom, I)
(would imagine)
(it doesn't really matter what you're)
(placing the bets on, red or black?)
(Like you don't really have to be betting)
(on the colour you lost on?) - No no no exactly.
Every time, because it's an equal chance
of each one coming up,
you you can switch. You could maybe go
red, black, red, black, red, it doesn't
matter because the probability will
always be a half
and and it will always be the same.
(So you don't have to bet on what you)
(lost on, you just have to- you just have to bet)
(again.) - Just have to bet again, yes exactly,
you can bet on anything -
in the 50/50 situation.
Another issue, nobody in the world has
unlimited money. It might feel like some
people do but there is a number
where the maximum amount any individual could bet.
That's our bank balance, n pounds. If at
any point
we need to bet more money than we
have we're screwed. Not only can we
not win our one pounds
we've literally lost everything that we own.
So we really don't want this to happen.
(Does your previous winnings)
(become part of that pot you can dig into?)
No so so the the 100 pound limit here
is the- like that's my pot. So even if
I start to win,
maybe I've won 50 in a row and now I in
theory have 150,
but what we're saying here is that 50
I've won that that's in a separate pot.
That's in like a-
that's in your money box, that goes in
the bank you cannot touch that.
We've said here that we're losing at
step k and in the next step
we would have to bet 2 to the power k. So
if I have to bet 2 to the k pounds;
if this is ever bigger than my
n pounds, game over.
Sad face, that's it. - (You're bankrupt)
We all cry. So
we can't have this happen. Now what you
can figure out is
how does this tell you about k?
Because k remember is the number of
losses that were allowed. So we can start
to plug some numbers in here and think
about, well if I go in with 100 pounds
how many losses am I allowed before I
lost all my hundred quid? You just
rearrange this slightly so we can say
2 to the k is equal to n, just to make
things a bit easier, so obviously if we
go a little bit above this
then we fail so let's look at the
boundary case of when they're equal.
So then you take log to the base 2 of
both sides
and what that does is tell you this
to be our value of k. Now this is quite
hard to interpret because we don't
really have any numbers,
so let's put some numbers in. So if I
come in with n
being 100 pounds. So if we start with 100 pounds
you're allowed seven losses, it's not too
bad. But it might seem unlikely,
I'm not gonna lose seven times in a
row, you would be amazed how common it is
to lose seven times in a row. Like there's
loads of really nice studies done on
this at how bad
we are as humans at thinking 'oh I'll
never be that unlucky'.
It's really easy to lose seven times in a row.
(Seven reds in a row and you're- you're out?)
If we're betting black yeah. And then
if you went up to say a thousand pounds
then you will get about 10 losses. So
even though you've
ten times your initial stake you only
get an extra three losses.
And this kind of pattern continues, it
starts to come down. So if you had 10,000
pounds
then you've got 14 losses and if you
even went all the way up to Brady's idea
of a million pounds
you've got 20 losses. You get twice as
many losses from 10 to 20 going from a
thousand to a million pounds. No matter
how much your money goes up this starts
to slow down quite drastically.
(And still this is only going to win you)
(one pound!) - This is to still win one pound,
yeah, this is just to win one pound. We
have this
relationship then between the number of
losses and the money that we've put in.
So what we can do then is say, right well,
it's a 50/50 game
so what is the probability that I
lose k times in a row? Well if I lose once
that's a half, then if I lose again
that's a half;
these are independent events it doesn't
matter what happened in the previous
one. So it's literally just a half to the
power k. So if I've lost
k times it's a half to the power k. This
is actually
1 over 2 to the k. And we have this
relationship up here
between 2 to the k and the amount of
money we start with, n.
So the probability that I lose k times
in a row and lose
all of my money and all of my belongings
is 1 divided by
the amount of money I have in the first
place. So perhaps as expected
the richer I am the less likely it is
that I lose all of my money. And we have
this really nice probability given in
terms of the amount of money we start with.
So if this is the probability that I lose
k times in a row, and lose all of my
fortune,
then the probability that I win -
which here is winning one pounds - is 1
minus this
because the total probability has to
always be 1. So the probability
that I win one pound using this strategy
is 1 minus 1 over n. And as n
increases this number goes really small
so the probability is really close to 1.
(If you Tom walk into the casino with a)
(hundred pounds in your back pocket, which)
(I think is conceivable-) - Okay yeah let's
go with that.
(And you play this strategy; how)
(likely is it you're gonna walk out with)
(your one pound profit?)
So if I put in, for n equals 100,
then the probability I win one pounds is
going to be
1 minus 1 over 100, so that's
going to be 99 out of 100. So there's a
99%
chance that I win a pound but, and this is
the key thing,
there's a 1% chance - really
small - that I lose 100 quid.
I really don't want to lose 100 quid to
have even- because I'm only doing it to
win a pound,
is it worth? So it's only a 1%
chance yes but is it worth that
1% chance that I lose my hundred
quid just to win a pound? If we do this
properly, I want to double
the money I start with. So we need to win
one pound 100 times in a row. So we need
to go through this whole strategy,
with our probability of 1 minus 1 over n,
but we need to do it
100 times in a row and so
I'm going to need more paper. The
probability of winning one pound
is 1 minus 1 over n. So the probability
of winning
100 times in a row, which means winning
100 pounds, is just-
well I need to win the first one, then I
need to win the second one,
so that's just multiply the probability
together, then I win the third - so it's
actually just this
to the power of a hundred. If I wanted to
win 100 times in a row it's this
probability
to a higher and higher power. We're using
the example here of me and my hundred pounds
if I just wanted the probability of
doubling my money,
for any n, then it's going to just be 1
minus 1 over n.
Each time I win one pound I've got to
win n times so it's 1 minus 1 over n
to the power n.
What is going on here? So let's put in
some numbers. If n is a hundred
we plug this into our formula, so it's
gonna be 1 minus 1 over 100
to the power 100, and that comes out to
be
0.366, so if we put that as a percent
it's going to be 36.6%. So a little bit
over a third
chance of me going in with my 100 pounds
and doubling my money through this
Martingale
doubling strategy. - (So this is a long night?)
This is a very long night yeah. Because that's-
I've literally got to win 100 times in a row.
So I have to employ this
strategy 100 times- and
and hope that none of them-
if I ever get a long
enough losing streak to run out of money
that's it I've lost my 100 pounds.
So I really do need to go back and
redo the same thing 100 times.
But if I do that I then got 36.6%
chance of doubling my money through this
strategy. Suppose I'm a little bit more
of a high roller and
I go for a thousand pounds, the value is
0.368.
So if I have a little bit more money I
get an increase,
0.2%. Still starting with one
dollar bet so I can never bet anything
more than a thousand,
that's like all I'm prepared to put forward
into this situation. And then if I was to
really go big
and start and say, well, you know what
I've got a million pounds of capital -
I wish - then the value here
is also 0.368. 36.8%.
So something's happening. This is this is-
as a mathematician this is where you get
your joy,
you sort of- you start thinking about
these fictional casinos, these
probability problems,
and then you start to spot patterns - what
the hell's going on?
What we're really doing here, as n is
increasing,
is you've got like a contest between
these two numbers.
Because as n gets bigger and bigger this
value in the bracket gets closer to 1
because you're taking away a smaller
smaller number. But as
n gets bigger you're taking a number
less than 1
to a larger and larger power. And when
you have numbers less than 1
to any power, so say a half squared, it
gets smaller.
So any number less than one raised to
any power,
positive whole number power, is going to
get smaller and smaller.
So this is always less than 1 and
we're raising it to a bigger and bigger power.
So it's- there's like a competition
because you're taking away smaller
pieces
but you're multiplying it by itself more
and more times. And that's why you get
this limit, it's an actual limiting
process, the two balance out and
eventually reach this constant. And the
constant,
and this is my favorite bit about this
whole problem, if you take
n to infinity and do a proper limiting
process on 1 minus 1 over n
to the n this is 1 over
e - euler's number, the natural rate of growth;
the number tattooed on my arm.
It's- it appears everywhere right?
Anything you're doing
with growth; it seems anything with
interest, gambling money, it's always e.
So here it's 1 over e. So the big
conclusion
is if I have a fixed amount of money,
n pounds, I'm willing to try and double
using the Martingale strategy, no matter
how much money I have
the absolute best chance I have of
doubling my money
is about 37%, or 1 over e. And there's
nothing I can do about that. If I get
more and more money
I still have the same chance, 36.8%
chance, of winning in this method. So
some of you may have spotted this
earlier or may realised what's going on
here.
And you have to think, well, we're playing
a 50/50 game,
like we're literally playing roulette
red or black. So we can go through all of this
and have, you know, the world's longest
stint in a casino trying to double a
million pounds
and do a million iterations of the
Martingale strategy
but I'm never going to get more than a
36.8% chance of doubling my money.
So why don't I just put everything on
black on the first spin?
Because then I've got a 50% chance of
actually
doubling my money. So the moral of the
story I think is
gambling strategies are all good and fun but
they're always gonna break down for many
reasons;
one of which is you don't have an
infinite amount of money. So even though
we can in theory win one pound every
time,
if you let time be long enough you're
never actually going to
to double your money or your your
expected value really does shrink to
zero.
So the Martingale strategy in theory
looks great,
if you do it with sensible numbers
saying I want to double my money,
literally you're better off just putting
it all on black and hoping for the best.
If you'd like to see even more
Numberphile videos with Tom Crawford,
well we've got a playlist. There's a link
on the screen and down in the video
description.
..Because there's more competition to eat
those rabbits. - Also Tom will be appearing
very soon on the Numberphile podcast so
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