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Gambling with the Martingale Strategy - Numberphile - YouTube
Channel: Numberphile
[0]
We're gonna go gambling. We're creating a-
[2]
let's call it a fictional casino
[4]
and we're playing a game with a 50/50
[8]
chance of winning.
[11]
Okay, so we're doing roulette -
red or black so
[13]
there's no zero, so it is
[14]
actually a 50/50 chance red or black.
[17]
What we're going to look at is something
[19]
called the Martingale Strategy.
[21]
Every time you lose a bet you double
[25]
down
[26]
on the next bet; and therefore eventually
you will
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win - is the idea.
- (Brad: What do you mean by)
[32]
(double down?)
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So if you bet one pound and lose then
[36]
you bet again
[37]
and you would bet two pounds; and then if
[39]
you lost again then you go four pounds;
[41]
if you lose you go eight pounds. So you
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double the last bet you made.
[46]
Provided you have pretty much unlimited
[48]
money you will eventually
[50]
win. A win is: come out of our fictional
[54]
casino with more money than I went in.
[55]
(You want to make a profit.)
- Exactly we're
[57]
making a profit.
- (Should we say we're)
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(always betting on red?)
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(Or black or-?)
- I think- I'm
[62]
tempted to go black.
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So I write with a black pen. I think if I
[66]
was given the choice I would- I would I'd
[68]
like a-
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I'm going for black over red I think.
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(Okay so we're betting on black.)
- Betting
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on black. And we're going to keep going;
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you think about well when is the first
[75]
time I lose? So if I lose at step k
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I've had my bet at step 1,
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and I bet one pound, and
then I must have lost
[83]
because we're assuming we're going to
[84]
keep going until this number k. Then at
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step 2 I bet two pounds. Then step three -
[90]
again we're assuming I've lost at step
[91]
2 - I double my bet
[93]
so this is now going to be four pounds.
[96]
Step 4 is going to be double that, eight
[98]
pounds, and this
[100]
carries on.
- (If you win your money gets)
[101]
(doubled?)
- If I win my money will be
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doubled, yes, it's a 50/50 game
[106]
such that whatever I bet- like in, exactly
[108]
like roulette. We don't have a zero
[110]
but the rules otherwise are the same. If
[112]
I bet on black, which is what we're doing,
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if it lands on black I double my money.
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(Because that zero is how
casino is insured.)
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The zero- exactly. You know this way.
The zero
[120]
is how- casinos know this and this is why
[122]
they have zero, because it's never a
[123]
50/50 chance.
[124]
But our mathematically fair casino gives
[127]
us a true 50/50 chance. At step
[129]
k then we can figure out how
much we would be
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betting because there's a pattern here.
[135]
So at step 1
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I bet one pounds, step 2 two pounds,
[139]
step 3 four pounds.
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So what's happening with our bets is
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these are increasing powers of 2.
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So this is times 2, times 2, times 2.
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So the actual formula
[150]
is going to be 2 to the power of k
[154]
minus 1. So we can just check this: so
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at step 1, so k is 1; 2 to the power
[160]
of 0 is 1.
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Step 2 k is 2, 2 to the power 1
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is 2;
[164]
2 squared is 4, 2 cubed is 8 so at
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step k we have just bet
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2 to the k minus 1 pounds.
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If I then lose at step k my total loss
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so far is the sum of all of these bets;
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so 1 plus 2
[182]
plus 4 plus 8 all the way up to 2
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to the k minus 1. We can write this as
[187]
the sum from
[188]
j equals 0 to k minus 1 of 2
[192]
power j. So 2 to the 0 is 1 then
[194]
plus
[195]
2 to the 1 is 2 plus 2 squared is
[197]
4. So this is a shorthand way of writing
[199]
out this sum.
[200]
And this is what we call a geometric
[202]
series; we
[203]
have a formula for the sum of a
[205]
geometric series so
[207]
we can plug it in. It's the
first term, which is
[210]
1, multiplied by 1 minus
[214]
the next term in the sequence. So it ends
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at 2 to the k minus 1
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so if we were to go to the next step
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that would be 2 to the k;
[224]
and then we divide by 1 minus
[227]
the common ratio between the terms.
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So what we're multiplying by at each
[233]
step, so divided by 1 minus 2. So you
[236]
tidy this up
[237]
and this is 2 to the k minus
[240]
1. So this is the total amount of money
[244]
I have lost when I lose my bet at step
[247]
k. So in the next step I'm going to bet
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2 to the k pounds - I'm going to win
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hopefully
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and then that covers my losses with one
[258]
pound to spare.
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So after all of this effort, after losing
[264]
at some step
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if I follow the Martingale strategy I
[268]
will eventually win -
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we hope - before I run out of money, that's
[273]
a whole other question,
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and I will win one pound. So we're not
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exactly-
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like it's good, it's a profit, but it's
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it's not like- we want more than a pound
[282]
don't we really? Like you don't really go
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to a casino to make a pound.
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(But if you scaled it to 10 pounds or a)
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(million pounds as)
[288]
(as your- like your multiple)
[291]
(you you're guaranteed to
win a million pounds!)
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Yeah, so so to win a million pounds
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you've got to
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go through this strategy one million
[298]
times, winning one pound each time.
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(Or just make that a million pound bet;)
[304]
(a two million pound bet; a four million)
[306]
(pound bet, all the way to a two)
[308]
(to the minus 1 times a million-)
- We
[310]
could also do that yeah. So we
could also change
[312]
the amount we bet at the beginning and
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then that would be-
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the amount that you your initial stake
[317]
is the amount that you'll win
[319]
through this strategy. So this strategy
[321]
is going to win you,
[323]
through this process, we're going to win
[324]
a pound each time. Now,
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there are problems with this. There is a
[329]
reason that this is not a foolproof
[331]
gambling strategy. First of all
[334]
most games aren't 50/50; if we're doing
[336]
roulette there's a zero which is green,
[339]
so it's not red or black, so you've not
[340]
actually got a 50/50 chance.
[342]
And that kind of really alters what's
[344]
what's happening.
- (In some casinos there)
[346]
(are two zeros)
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Yes zero and double zero, exactly yeah, so
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they're even- they're even
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stingier those odds. The other thing is
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casinos have maximum bet limits.
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In most casinos you can't walk in and
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just drop a million pounds,
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like they're gonna be like no there's a
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maximum bet limit here.
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And even if you start really low and you
[364]
keep losing
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this will grow really quickly. This is
[367]
exponential growth which
[369]
we know a lot about these days. And this
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will grow really quickly
[372]
and it will actually reach a million a
[373]
lot faster than you realise.
- (So if red comes up)
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(lots of times in a row - ?)
- Yeah you're going
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to reach the maximum betting limit in an
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actual casino and then
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you're screwed, you've then lost like a
[382]
million pounds, you've not got a chance
[384]
to get your money back.
[385]
Because that's the key; if this fails
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you've lost
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2 to the power k minus 1 pounds which is
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a lot of money if
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k is like 10 or something, like, so it can
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really really ramp up.
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(Just for the sake of clarity too Tom, I)
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(would imagine)
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(it doesn't really matter what you're)
[402]
(placing the bets on, red or black?)
[404]
(Like you don't really have to be betting)
[405]
(on the colour you lost on?)
- No no no exactly.
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Every time, because it's an equal chance
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of each one coming up,
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you you can switch. You could maybe go
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red, black, red, black, red, it doesn't
[414]
matter because the probability will
[415]
always be a half
[416]
and and it will always be the same.
[418]
(So you don't have to bet on what you)
[419]
(lost on, you just have to- you just
have to bet)
[421]
(again.)
- Just have to bet again, yes exactly,
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you can bet on anything -
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in the 50/50 situation.
[430]
Another issue, nobody in the world has
[432]
unlimited money. It might feel like some
[433]
people do but there is a number
[435]
where the maximum amount
any individual could bet.
[439]
That's our bank balance, n pounds. If at
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any point
[442]
we need to bet more money than we
[446]
have we're screwed. Not only can we
[450]
not win our one pounds
[451]
we've literally lost
everything that we own.
[455]
So we really don't want this to happen.
[457]
(Does your previous winnings)
[460]
(become part of that
pot you can dig into?)
[463]
No so so the the 100 pound limit here
[467]
is the- like that's my pot. So even if
[470]
I start to win,
[471]
maybe I've won 50 in a row and now I in
[474]
theory have 150,
[476]
but what we're saying here is that 50
[478]
I've won that that's in a separate pot.
[480]
That's in like a-
[481]
that's in your money box, that goes in
[482]
the bank you cannot touch that.
[484]
We've said here that we're losing at
[486]
step k and in the next step
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we would have to bet 2 to the power k. So
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if I have to bet 2 to the k pounds;
[495]
if this is ever bigger than my
[498]
n pounds, game over.
[502]
Sad face, that's it.
- (You're bankrupt)
[504]
We all cry. So
[506]
we can't have this happen. Now what you
[507]
can figure out is
[509]
how does this tell you about k?
[512]
Because k remember is the number of
[514]
losses that were allowed. So we can start
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to plug some numbers in here and think
[518]
about, well if I go in with 100 pounds
[519]
how many losses am I allowed before I
[521]
lost all my hundred quid? You just
[522]
rearrange this slightly so we can say
[524]
2 to the k is equal to n, just to make
[527]
things a bit easier, so obviously if we
[528]
go a little bit above this
[529]
then we fail so let's look at the
[531]
boundary case of when they're equal.
[532]
So then you take log to the base 2 of
[535]
both sides
[536]
and what that does is tell you this
[539]
to be our value of k. Now this is quite
[541]
hard to interpret because we don't
[542]
really have any numbers,
[543]
so let's put some numbers in. So if I
[545]
come in with n
[546]
being 100 pounds. So if we
start with 100 pounds
[549]
you're allowed seven losses, it's not too
[551]
bad. But it might seem unlikely,
[553]
I'm not gonna lose seven times in a
[555]
row, you would be amazed how common it is
[558]
to lose seven times in a row. Like there's
[560]
loads of really nice studies done on
[561]
this at how bad
[563]
we are as humans at thinking 'oh I'll
[565]
never be that unlucky'.
[567]
It's really easy to lose
seven times in a row.
[568]
(Seven reds in a row and
you're- you're out?)
[570]
If we're betting black yeah. And then
[571]
if you went up to say a thousand pounds
[574]
then you will get about 10 losses. So
[577]
even though you've
[578]
ten times your initial stake you only
[580]
get an extra three losses.
[581]
And this kind of pattern continues, it
[583]
starts to come down. So if you had 10,000
[585]
pounds
[586]
then you've got 14 losses and if you
[590]
even went all the way up to Brady's idea
[593]
of a million pounds
[594]
you've got 20 losses. You get twice as
[596]
many losses from 10 to 20 going from a
[598]
thousand to a million pounds. No matter
[600]
how much your money goes up this starts
[602]
to slow down quite drastically.
[604]
(And still this is only going to win you)
[605]
(one pound!)
- This is to still win one pound,
[608]
yeah, this is just to win one pound. We
[610]
have this
[611]
relationship then between the number of
[612]
losses and the money that we've put in.
[614]
So what we can do then is say, right well,
[616]
it's a 50/50 game
[618]
so what is the probability that I
[621]
lose k times in a row? Well if I lose once
[625]
that's a half, then if I lose again
[627]
that's a half;
[628]
these are independent events it doesn't
[630]
matter what happened in the previous
[632]
one. So it's literally just a half to the
[634]
power k. So if I've lost
[636]
k times it's a half to the power k. This
[638]
is actually
[639]
1 over 2 to the k. And we have this
[642]
relationship up here
[643]
between 2 to the k and the amount of
[645]
money we start with, n.
[646]
So the probability that I lose k times
[649]
in a row and lose
[650]
all of my money and all of my belongings
[652]
is 1 divided by
[655]
the amount of money I have in the first
[656]
place. So perhaps as expected
[659]
the richer I am the less likely it is
[663]
that I lose all of my money. And we have
[665]
this really nice probability given in
[667]
terms of the amount of
money we start with.
[669]
So if this is the probability that I lose
[672]
k times in a row, and lose all of my
[675]
fortune,
[676]
then the probability that I win -
[680]
which here is winning one pounds - is 1
[682]
minus this
[684]
because the total probability has to
[685]
always be 1. So the probability
[687]
that I win one pound using this strategy
[690]
is 1 minus 1 over n. And as n
[694]
increases this number goes really small
[697]
so the probability is really close to 1.
[699]
(If you Tom walk into the casino with a)
[701]
(hundred pounds in
your back pocket, which)
[703]
(I think is conceivable-)
- Okay yeah let's
[705]
go with that.
[706]
(And you play this strategy; how)
[708]
(likely is it you're gonna walk out with)
[709]
(your one pound profit?)
[710]
So if I put in, for n equals 100,
[714]
then the probability I win one pounds is
[716]
going to be
[717]
1 minus 1 over 100, so that's
[720]
going to be 99 out of 100. So there's a
[722]
99%
[723]
chance that I win a pound but, and this is
[726]
the key thing,
[727]
there's a 1% chance - really
[729]
small - that I lose 100 quid.
[732]
I really don't want to lose 100 quid to
[734]
have even- because I'm only doing it to
[736]
win a pound,
[737]
is it worth? So it's only a 1%
[738]
chance yes but is it worth that
[741]
1% chance that I lose my hundred
[743]
quid just to win a pound? If we do this
[745]
properly, I want to double
[747]
the money I start with. So we need to win
[751]
one pound 100 times in a row. So we need
[754]
to go through this whole strategy,
[756]
with our probability of 1 minus 1 over n,
[758]
but we need to do it
[760]
100 times in a row and so
[763]
I'm going to need more paper. The
[765]
probability of winning one pound
[766]
is 1 minus 1 over n. So the probability
[769]
of winning
[770]
100 times in a row, which means winning
[772]
100 pounds, is just-
[774]
well I need to win the first one, then I
[776]
need to win the second one,
[777]
so that's just multiply the probability
[779]
together, then I win the third - so it's
[781]
actually just this
[782]
to the power of a hundred. If I wanted to
[785]
win 100 times in a row it's this
[787]
probability
[787]
to a higher and higher power. We're using
[789]
the example here of me
and my hundred pounds
[791]
if I just wanted the probability of
[794]
doubling my money,
[796]
for any n, then it's going to just be 1
[799]
minus 1 over n.
[800]
Each time I win one pound I've got to
[802]
win n times so it's 1 minus 1 over n
[805]
to the power n.
[805]
What is going on here? So let's put in
[808]
some numbers. If n is a hundred
[809]
we plug this into our formula, so it's
[811]
gonna be 1 minus 1 over 100
[813]
to the power 100, and that comes out to
[816]
be
[818]
0.366, so if we put that as a percent
[821]
it's going to be 36.6%. So a little bit
[824]
over a third
[825]
chance of me going in with my 100 pounds
[828]
and doubling my money through this
[830]
Martingale
[831]
doubling strategy.
- (So this is a long night?)
[834]
This is a very long night
yeah. Because that's-
[836]
I've literally got to win
100 times in a row.
[838]
So I have to employ this
[839]
strategy 100 times- and
[842]
and hope that none of them-
[846]
if I ever get a long
[847]
enough losing streak to run out of money
[849]
that's it I've lost my 100 pounds.
[850]
So I really do need to go back and
[852]
redo the same thing 100 times.
[855]
But if I do that I then got 36.6%
[858]
chance of doubling my money through this
[860]
strategy. Suppose I'm a little bit more
[862]
of a high roller and
[863]
I go for a thousand pounds, the value is
[867]
0.368.
[869]
So if I have a little bit more money I
[871]
get an increase,
[872]
0.2%. Still starting with one
[874]
dollar bet so I can never bet anything
[875]
more than a thousand,
[876]
that's like all I'm prepared
to put forward
[878]
into this situation. And then if I was to
[880]
really go big
[880]
and start and say, well, you know what
[882]
I've got a million pounds of capital -
[884]
I wish - then the value here
[887]
is also 0.368. 36.8%.
[892]
So something's happening. This is this is-
[895]
as a mathematician this is where you get
[896]
your joy,
[897]
you sort of- you start thinking about
[899]
these fictional casinos, these
[900]
probability problems,
[902]
and then you start to spot patterns - what
[904]
the hell's going on?
[905]
What we're really doing here, as n is
[907]
increasing,
[909]
is you've got like a contest between
[912]
these two numbers.
[913]
Because as n gets bigger and bigger this
[916]
value in the bracket gets closer to 1
[918]
because you're taking away a smaller
[920]
smaller number. But as
[922]
n gets bigger you're taking a number
[925]
less than 1
[926]
to a larger and larger power. And when
[929]
you have numbers less than 1
[931]
to any power, so say a half squared, it
[934]
gets smaller.
[935]
So any number less than one raised to
[938]
any power,
[938]
positive whole number power, is going to
[941]
get smaller and smaller.
[942]
So this is always less than 1 and
[944]
we're raising it to a
bigger and bigger power.
[945]
So it's- there's like a competition
[948]
because you're taking away smaller
[949]
pieces
[949]
but you're multiplying it by itself more
[951]
and more times. And that's why you get
[954]
this limit, it's an actual limiting
[956]
process, the two balance out and
[958]
eventually reach this constant. And the
[959]
constant,
[960]
and this is my favorite bit about this
[962]
whole problem, if you take
[963]
n to infinity and do a proper limiting
[966]
process on 1 minus 1 over n
[969]
to the n this is 1 over
[973]
e - euler's number, the
natural rate of growth;
[977]
the number tattooed on my arm.
[980]
It's- it appears everywhere right?
[982]
Anything you're doing
[983]
with growth; it seems anything with
[985]
interest, gambling money, it's always e.
[987]
So here it's 1 over e. So the big
[990]
conclusion
[992]
is if I have a fixed amount of money,
[996]
n pounds, I'm willing to try and double
[999]
using the Martingale strategy, no matter
[1002]
how much money I have
[1005]
the absolute best chance I have of
[1007]
doubling my money
[1008]
is about 37%, or 1 over e. And there's
[1011]
nothing I can do about that. If I get
[1013]
more and more money
[1013]
I still have the same chance, 36.8%
[1017]
chance, of winning in this method. So
[1020]
some of you may have spotted this
[1023]
earlier or may realised what's going on
[1024]
here.
[1025]
And you have to think, well, we're playing
[1028]
a 50/50 game,
[1029]
like we're literally playing roulette
[1030]
red or black. So we can go
through all of this
[1034]
and have, you know, the world's longest
[1036]
stint in a casino trying to double a
[1037]
million pounds
[1039]
and do a million iterations of the
[1041]
Martingale strategy
[1044]
but I'm never going to get more than a
[1045]
36.8% chance of doubling my money.
[1048]
So why don't I just put everything on
[1050]
black on the first spin?
[1053]
Because then I've got a 50% chance of
[1056]
actually
[1057]
doubling my money. So the moral of the
[1059]
story I think is
[1062]
gambling strategies are
all good and fun but
[1065]
they're always gonna break down for many
[1068]
reasons;
[1069]
one of which is you don't have an
[1071]
infinite amount of money. So even though
[1073]
we can in theory win one pound every
[1075]
time,
[1076]
if you let time be long enough you're
[1078]
never actually going to
[1079]
to double your money or your your
[1080]
expected value really does shrink to
[1082]
zero.
[1083]
So the Martingale strategy in theory
[1086]
looks great,
[1087]
if you do it with sensible numbers
[1089]
saying I want to double my money,
[1091]
literally you're better off just putting
[1092]
it all on black and hoping for the best.
[1096]
If you'd like to see even more
[1098]
Numberphile videos with Tom Crawford,
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well we've got a playlist. There's a link
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on the screen and down in the video
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description.
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..Because there's more competition to eat
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those rabbits.
- Also Tom will be appearing
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very soon on the Numberphile podcast so
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keep an ear out for that.
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Thank you also to Numberphile's patreon
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supporters.
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We have a list of the whole list of
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supporters, I'll include a link to that
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in the video description; but a few of
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them are also listed here on the screen
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you can see their names at the moment.
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Special thanks to them. If you'd like to
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help us out go to
patreon.com/numberphile.
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We really appreciate it
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but we also appreciate everyone who just
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takes the time to watch and like and
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share these videos,
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and that includes you. Yes you, I'm
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talking to you specifically.
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