## Computing variance for an individual blackjack hand

The variance of an individual blackjack hand that does not involve splitting a pair can be computed from the parameters listed below and their relationships. Below var refers to variance of a given player hand for a given strategy versus a given player up card and vari refers to variance for hand and strategy versus dealer up card of ace given player buys insurance. In order to compare var for a given hand and strategy to vari it should be computed versus up card of ace while vari is computed for the same given hand and strategy (v ace.)

ev is unconditional expected value for a given strategy meaning a non-blackjack loses to dealer blackjack.

## Parameters

pWin = probability of winning the hand

pLose = probability of losing the hand (includes loss to dealer blackjack)

pLose_noDBJ = probability of losing hand where loss is not due to dealer blackjack

pTie = probability of pushing the hand

pDBJ = probability of dealer blackjack

bjOdds = odds paid for a player blackjack

ev = expected value of the hand for a given player strategy versus given dealer up card

evi = expected value of the hand for a given player strategy versus up card of ace given player buys insurance

var = variance (no insurance)

vari = variance (insurance taken)

### Undoubled player blackjack

pWin + pLose + pTie = 1

pLose = 0

pTie = pDBJ

pWin = 1 - pTie

ev(stand v ace) = (1 - pTie)*bjOdds = (1 - pDBJ)*bjOdds

var = pTie*(0 - ev)^2 + pWin*(bjOdds - ev)^2

var = ev^2 + (1 - pTie)*(bjOdds^2 - 2*bjOdds*ev) can be algebraically derived from above info

evi = bjOdds*(1 - pDBJ) + .5*(3*pDBJ - 1)

vari = pDBJ*(1 - evi)^2 + (1 - pDBJ)*((bjOdds - .5) - evi)^2

special case example:

__bjOdds = 1.5 and pDBJ = 1/3__

applies to any composition where true

ev = evi = 1

var = .5

vari = 0

### Undoubled hand (non-blackjack)

pWin + pLose + pTie = 1

pWin - pLose = ev

pWin = (1 - pTie + ev)/2

pLose = (1 - pTie - ev)/2

var = pLose*(-1 - ev)^2 + pTie*(0 - ev)^2 + pWin*(1 - ev)^2

var = 1 - pTie - ev^2 can be algebraically derived from above info

evi = ev(up card ace) + .5*(3*pDBJ - 1)

pLose_noDBJ = pLose - pDBJ

vari = pDBJ*(0 - evi)^2 + pLose_noDBJ*(-1.5 - evi)^2 + pTie*(-.5 - evi)^2 + pWin*(.5 - evi)^2

special case examples:

__9-7 v ace single deck s17__

composition before hand: {4,3,4,4,4,4,4,4,4,16}

composition after hand given up card ace: {3,3,4,4,4,4,3,4,3,16}

ev(hit v ace): -0.67000

pTie, pWin, pLose (hit v ace): 0.03944, 0.14528, 0.81528

var: 0.51166

pDBJ: 16/48 = 1/3

evi = ev(hit v ace) = -0.67000

pLose_noDBJ: 0.48194

vari: 0.68166

__T-T v ace single deck s17__

composition before hand: {4,3,3,3,3,3,3,3,4,16}

composition after hand given up card ace: {3,3,3,3,3,3,3,3,4,14}

ev(stand v ace): 0.09280

pTie, pWin, pLose (stand v ace): 0.14279, 0.47501, 0.38204

var: 0.84860

pDBJ: 14/42 = 1/3

evi = ev(stand v ace): 0.09280

pLose_noDBJ: 0.04887

vari: 0.25580

### Doubled hand (ENHC)

pWin + pLose + pTie = 1

2*(pWin - pLose) = ev

pWin = (2*(1 - pTie) + ev)/4

pLose = (2*(1 - pTie) - ev)/4

var = pWin*(2 - ev)^2 + pLose*(-2 - ev)^2 + pTie*(0 - ev)^2

var = 4*(1 - pTie) - ev^2 can be algebraically derived from above info

evi = ev(up card ace) + .5*(3*pDBJ - 1)

pLose_noDBJ = pLose - pDBJ

vari = pDBJ*(0 - evi)^2 + pLose_noDBJ*(-2.5 - evi)^2 + pTie*(-.5 - evi)^2 + pWin*(1.5 - evi)^2

special case example:

__6-5 v ace single deck s17__

composition before hand: {4,3,4,4,4,4,4,4,4,16}

composition after hand given up card ace: {3,3,4,4,3,3,4,4,4,16}

ev(dbl v ace): -0.48724

pTie, pLose, pWin (dbl v ace): 0.05024, 0.51336, 0.43640

var: 3.56161

pDBJ: 16/48 = 1/3

evi = ev(dbl v ace): -0.48724

pLose_noDBJ: 0.18002

vari: 2.53185

### Doubled hand (Full peek)

pWin + pLose + pTie = 1

2*(pWin - pLose) + pDBJ = ev

pWin = (2*(1 - pTie) + ev)/4

pLose = (2*(1 - pTie) - ev)/4

pLose_noDBJ = pLose - pDBJ

var = pWin*(2 - ev)^2 + pLose_noDBJ*(-2 - ev)^2 + pTie*(0 - ev)^2 + pDBJ*(-1 - ev)^2

var = 4*(1 - pTie) - 3*pDBJ - ev^2 can be algebraically derived from above info

evi = ev(up card ace) + .5*(3*pDBJ - 1)

vari = pDBJ*(0 - evi)^2 + pLose_noDBJ*(-2.5 - evi)^2 + pTie*(-.5 - evi)^2 + pWin*(1.5 - evi)^2

special case example:

__6-5 v ace single deck s17__

composition before hand: {4,3,4,4,4,4,4,4,4,16}

composition after hand given up card ace: {3,3,4,4,3,3,4,4,4,16}

ev(dbl v ace): -0.15391

pTie, pLose, pWin (dbl v ace): 0.05024, 0.51336, 0.43640

var: 2.77533

pDBJ: 16/48 = 1/3

evi = ev(dbl v ace): -0.15391

pLose_noDBJ: 0.18002

vari: 2.19851

## Computing variance for pair split

The variance of a pair split can also be computed but is more complicated. If number of allowed splits is 1 then the range of units won/lost is -2 to +2 if there is no doubling after splitting and the range is -4 to +4 if doubling after splitting is allowed. Each additional allowed split increases this range. To compute variance the probability of each possible units won/lost needs to be computed. The problem with this is that as number of decks and number of allowed splits increases the number of required iterations grows exponentially. For each necessary iteration dealer probabilites need to be calculated. I am able to get values for single deck, 1 allowed split which takes a while and 2 decks, 1 allowed split which takes even longer.