standard deviation formula question

Looking through some old threads I found a post by DrEntropy:

"For a wager that pays x:1 with a probability of winning p, the house edge and standard deviation (per unit bet and per 'root decision') are:
ev = (x+1)*p - 1
std = (x+1)*Sqrt[p*(1-p)]"

I understand that for the pass line bet in craps you get: (1+1)(Sqrt(.5071)(.4929))=.9994

And for an even money roulette wager: (1+1)(Sqrt(.5263)(.4737))=.9986

A second way of getting .9986 is (9/19)(1-(-.0526))^2 + (10/19)(-1-(-.0526))^2 =.9972, Sqrt (.9972) = .9986

My question is how do you use this second way (if you can) to get .9994 for the pass line bet?

The probabilty of winning a pass line bet is 244/495, and the probability of losing is 251/495. The ev of the pass line bet is 2*244/495 - 1 = -7/495.

The variance of the pass line bet is (244/495)*[ +1 - (-7/495) ]^2 + (251/495)*[ -1 - (-7/495) ]^2, which is about 0.9998. The standard deviation is the square root of the variance and is about 0.9999 for the pass line bet.

Quote: mikey41

And for an even money roulette wager: (1+1)(Sqrt(.5263)(.4737))=.9986

A second way of getting .9986 is (9/19)(1-(-.0526))^2 + (10/19)(-1-(-.0526))^2 =.9972, Sqrt (.9972) = .9986

FYI, The variance is real easy using the binomial$stdev formula you found
just
var = (x+1)^2 *p*(1-p)

Quote: mikey41

My question is how do you use this second way (if you can) to get .9994 for the pass line bet?

the house edge for the pass line is -7/495
(x+1)*p - 1
2*(244/495) - 1 = yes

plug that in and see if that works (it should as another posted)

that method looks to take the long way to get home for simple binomial type bets

Sally

Quote: mikey41

My question is how do you use this second way (if you can) to get .9994 for the pass line bet?

You realize that both ways are the same? The first formula is derived from the second one which is simply the definition of SD:
Variance of binomial distribution

Thanks, ChesterDog. That clears up everything. I don't know what I was doing wrong before. I thought I input the same numbers you showed (although I used decimal approximations) in the calculator and kept getting the wrong number for the variance.

Quote: mustangsally

that method looks to take the long way to get home for simple binomial type bets

Sally

I know. I didn't understand how the formulas were related as Paisiello indicated. Now it all makes sense. Thanks again everyone.