Don,
Suppose someone receives $1500 if a fair coin comes up heads and loses $500 if it lands tails. EV is $500/toss. What is the variance per toss and how do you calculate it? If it is not too much trouble please walk me through it.
My other question pertains to average win and loss in blackjack. Is there a method to estimate average win/loss for a counter given his hourly EV, SD, and Variance? Thanks for any answers.
MJ
"Suppose someone receives $1500 if a fair coin comes up heads and loses $500 if it lands tails. EV is $500/toss. What is the variance per toss"
$1,000,000.
"and how do you calculate it? If it is not too much trouble please walk me through it."
The formula for the variance of a discrete random variable is E[X^2] - E[X]^2. In words, it is the expected squared value, minus the square of the expected value. In this example, each result happens with 50% probability, so the average, or expected, squared value is simply: [(1500)^2 + (-500)^2]/2 = 1,250,000.
The expected value, E[X], as you stated, is just (1500-500)/2 = 500. So, E[X]^2 = 500^2 = 250,000.
So variance = 1,250,000 - 250,000 = 1,000,000.
You didn't ask, but, since s.d. is the square root of variance, the s.d. per toss would be 1,000.
"My other question pertains to average win and loss in blackjack. Is there a method to estimate average win/loss for a counter given his hourly EV, SD, and Variance? Thanks for any answers."
Yes. VERY complicated formula. You can find a detailed discussion of the entire phenomenon (only one I've ever seen of it), with examples, on pp. 298-301 of BJA3.
Hope this helps.
Don
Thank you for the meticulous explanation. Suppose one wanted to find the variance for x tosses. Is it equal to 1000000x?
MJ
As DH has indicated, variances can be added linearly. That's one of the beauties of working with variance instead of s.d. The latter can't be added linearly. Need to do a root-mean-square calculation, instead.
Don
......but I do have BJA1 and BJA2.
Suppose someone receives $1500 if a fair coin comes up heads and loses $500 if it lands tails. EV is $500/toss.
Wouldn't the expected value per toss be $1,000. That is the $1,500 possible win minus the $500 possible loss.
What is the variance per toss
How can it be $1,000,000 if the most one can win per toss is $1,500 and the most one can loss per toss is $500.
"......but I do have BJA1 and BJA2."
Shame on you! :-) BJA1 + BJA2 ≠ BJA3!!
"Suppose someone receives $1500 if a fair coin comes up heads and loses $500 if it lands tails. EV is $500/toss.
"Wouldn't the expected value per toss be $1,000. That is the $1,500 possible win minus the $500 possible loss."
Er, no. You forgot to divide by two. 50% chance to win $1,500 = +$750 e.v. 50% chance to LOSE $500 = -$250 e.v. Added, that equals +$500.
"What is the variance per toss
"How can it be $1,000,000 if the most one can win per toss is $1,500 and the most one can loss per toss is $500?"
Because that's the formula for variance! Maybe you're thinking of standard deviation, which, in this case, is $1,000. Variance is the SQUARE of s.d.
Don
