Math Is Fun Forum

jacobite

Member

Registered: 2016-01-14

Posts: 2

Expectation/Probability problem

Hi folks,

I'm trying to solve a problem on expectation and the answer I'm getting isn't what I think it should be; can anyone see where I'm making a mistake?

The problem concerns roulette where the probability of a win on the even chances is 18/37 and a loss is 1 - 18/37 = 19/37. Roulette has a negative mathematical expectation of -1/37 and all expectations of various bets should result in this.

Suppose I bet on Red for two spins and use a betting scheme where I raise the bet from $1 to $2 after a loss. The possibilities are (W = win and L = loss):

WW probability: (18/37)^2 gain: +$2
WL probability: (18/37)*(19/37) gain: $0
LW probability: (19/37)*(18/37) gain: +$1 (-$1, then bet $2 and win)
LL probability: (19/37)^2 gain: -$3 (-$1, then bet $2 and lose)

The expectation is

E(X) = (18/37)^2*2 + (18/37)*(19/37)*0 + (19/37)*(18/37)*1 + (19/37)^2*(-3) = -0.0697

But it should be -0.027 (equivalent to -1/37)

Thanks in advance.

Last edited by jacobite (2016-01-15 00:48:37)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Expectation/Probability problem

Hi;

That E(x) means what you would be stuck betting 1 and 2 units in your scheme. So you are stuck .0697 units. The more you play the lower this is going to get, because you are going to lose more.

In betting terms it means you are down almost 7 cents, if your units are 1 dollar and 2 dollars.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

jacobite

Member

Registered: 2016-01-14

Posts: 2

Re: Expectation/Probability problem

Hi bobbym,

Thanks for the reply. I must admit I'm a bit confused now. Perhaps what I need to do is divide by the average stake because the -0.027 applies to the amount returned PER UNIT staked? In that case I'm unsure what the average stake is in this case.

But then, thinking about it that can't be right because the average stake would be between $1 and $2, and to get -0.027 from -0.0697 I would need to divide by about 2.5.

Last edited by jacobite (2016-01-15 04:05:40)

Relentless

Member

Registered: 2015-12-15

Posts: 631

Re: Expectation/Probability problem

Hi,

It would actually be -2/37 if you only bet $1 (since you are betting twice), which is about -0.054 and much closer. The change to -0.0679 is due to the conditional $2 bet; since you stake more, expectation goes down.

Specifically, the value of LL drops by about -0.264, while the value of LW rises by about 0.2498

Last edited by Relentless (2016-01-17 05:07:41)