I got the following info from wizard of odds website... some of it may help you with your calculations.
you may need to go to his site and use his house edge calculator for each blackjack game you play depending on house rules (#of decks, # of splits...)
also since a lot of online games shuffle more often (some even after each hand) you may want to read: The effect on the house edge of the continuous shuffling machines vs. the cut card. which is located on his site.
good luck:
I played t hands of blackjack one at a time, flat betting and religiously following basic strategy. My final outcome was a gain of r (a negative r represents a loss). The house edge under the rules I played was h. What is the probability of losing this much or more in a fair game?
Using Excel type this into any cell, substituting the correct values for h, t, and r:
=normsdist((r+t*h+0.5)/(t^0.5*1.16)).
The 1.16 is the standard deviation per hand. This will vary slightly from one set of rule to another but I feel 1.16 is a good benchmark. Let's take an example. Suppose the number of hands is 1000, the house edge is 0.41%, and the player lost 100 units. The formula for the probability of losing this many units or more is =normsdist((-100+1000*0.0041+0.5)/(1000^0.5*1.16)) = normsdist(-2.600700765) = 0.004651712 .
This appendix presents information pertinent to the standard deviation in blackjack. It should be noted my numbers come in a little higher than those using Stanford Wong's Blackjack Count Analyzer. My standard deviations come in about 0.016 higher. That said, the following table indicates the possible outcomes based on a 200 million hand simulation. Atlantic City rules (8 decks, double any first two cards, double after a split, split up to four hands, no surrender, dealer stands on soft 17) were used.
<CENTER><TABLE cellSpacing=2 cellPadding=1 bgColor=#e0d18b border=1><TBODY><TR><TD align=middle colSpan=3>Final outcome in blackjack</TD></TR><TR><TD align=middle>Net win</TD><TD align=middle>Number</TD><TD align=middle>Probability</TD></TR><TR><TD align=right>-8</TD><TD align=right>48</TD><TD align=right>0.00000024</TD></TR><TR><TD align=right>-7</TD><TD align=right>653</TD><TD align=right>0.00000327</TD></TR><TR><TD align=right>-6</TD><TD align=right>4654</TD><TD align=right>0.00002327</TD></TR><TR><TD align=right>-5</TD><TD align=right>22415</TD><TD align=right>0.00011208</TD></TR><TR><TD align=right>-4</TD><TD align=right>114254</TD><TD align=right>0.00057127</TD></TR><TR><TD align=right>-3</TD><TD align=right>465229</TD><TD align=right>0.00232615</TD></TR><TR><TD align=right>-2</TD><TD align=right>9248859</TD><TD align=right>0.04624430</TD></TR><TR><TD align=right>-1</TD><TD align=right>85919012</TD><TD align=right>0.42959500</TD></TR><TR><TD align=right>0</TD><TD align=right>17594957</TD><TD align=right>0.08797480</TD></TR><TR><TD align=right>1</TD><TD align=right>64221574</TD><TD align=right>0.32110800</TD></TR><TR><TD align=right>1.5</TD><TD align=right>9052402</TD><TD align=right>0.04526200</TD></TR><TR><TD align=right>2</TD><TD align=right>12616269</TD><TD align=right>0.06308130</TD></TR><TR><TD align=right>3</TD><TD align=right>525109</TD><TD align=right>0.00262555</TD></TR><TR><TD align=right>4</TD><TD align=right>169229</TD><TD align=right>0.00084615</TD></TR><TR><TD align=right>5</TD><TD align=right>33994</TD><TD align=right>0.00016997</TD></TR><TR><TD align=right>6</TD><TD align=right>9610</TD><TD align=right>0.00004805</TD></TR><TR><TD align=right>7</TD><TD align=right>1576</TD><TD align=right>0.00000788</TD></TR><TR><TD align=right>8</TD><TD align=right>156</TD><TD align=right>0.00000078</TD></TR></TBODY></TABLE></CENTER>
The table shows the following information:
- Probability of net win = 43.31%
- Probability of tie = 8.80%
- Probability of net loss = 47.89%
- Average loss = 1.11
- Average win = 1.22
- Standard deviation = 1.17
The next table shows the stardard variation under other rule variations. Note that the most important variable is whether the player is allowed to double on soft totals, followed by being allowed to double after a split.
<CENTER><TABLE cellSpacing=2 cellPadding=1 bgColor=#e0d18b border=1><TBODY><TR><TD align=middle colSpan=5>Standard Deviation in Blackjack</TD></TR><TR><TD align=middle>Number
of Decks</TD><TD align=middle>Maximum
Splits</TD><TD align=middle>Double
after
Split</TD><TD align=middle>Soft
Doubling
Allowed</TD><TD align=middle>Standard
Deviation</TD></TR><TR><TD align=right>1</TD><TD align=right>1</TD><TD align=middle>No</TD><TD align=middle>Yes</TD><TD align=right>1.1450</TD></TR><TR><TD align=right>1</TD><TD align=right>2</TD><TD align=middle>Yes</TD><TD align=middle>Yes</TD><TD align=right>1.1582</TD></TR><TR><TD align=right>2</TD><TD align=right>2</TD><TD align=middle>Yes</TD><TD align=middle>Yes</TD><TD align=right>1.1638</TD></TR><TR><TD align=right>4</TD><TD align=right>2</TD><TD align=middle>Yes</TD><TD align=middle>Yes</TD><TD align=right>1.1666</TD></TR><TR><TD align=right>6</TD><TD align=right>2</TD><TD align=middle>Yes</TD><TD align=middle>Yes</TD><TD align=right>1.1676</TD></TR><TR><TD align=right>8</TD><TD align=right>1</TD><TD align=middle>Yes</TD><TD align=middle>Yes</TD><TD align=right>1.1611</TD></TR><TR><TD align=right>8</TD><TD align=right>2</TD><TD align=middle>Yes</TD><TD align=middle>Yes</TD><TD align=right>1.1680</TD></TR><TR><TD align=right>8</TD><TD align=right>3</TD><TD align=middle>Yes</TD><TD align=middle>Yes</TD><TD align=right>1.1695</TD></TR><TR><TD align=right>8</TD><TD align=right>3</TD><TD align=middle>No</TD><TD align=middle>Yes</TD><TD align=right>1.1468</TD></TR><TR><TD align=right>8</TD><TD align=right>3</TD><TD align=middle>Yes</TD><TD align=middle>No</TD><TD align=right>1.1417</TD></TR></TBODY></TABLE></CENTER>
Following is the standard deviation by playing 1 to 3 hands at the same time, the same amount bet on each hand. These numbers are courtesy of Marsha Ness, who used Blackjack Audit simulation software. The rules are standard Atlantic City: 6 decks, dealer stands on soft 17, double any two cards, double after split, no surrender, split up to 4 hands, no resplitting aces, no drawing to split aces.
<CENTER><TABLE cellSpacing=2 cellPadding=1 bgColor=#e0d18b border=1><TBODY><TR><TD align=middle colSpan=2>Standard Deviation by Number of Hands</TD></TR><TR><TD align=middle>Number
of Hands</TD><TD align=middle>Standard
Deviation</TD></TR><TR><TD align=right>1</TD><TD align=right>1.15514</TD></TR><TR><TD align=right>2</TD><TD align=right>1.90837</TD></TR><TR><TD align=right>3</TD><TD align=right>2.63198</TD></TR></TBODY></TABLE></CENTER>
On the same topic according to
Professional Blackjack by Stanford Wong (page 203) the variance for similar rules is 1.32 and the covariance is 0.48. The total variance of n hands would be 1.32*n + 0.48*n*(n-1). Take the final square root to get the standard deviation.
The next table is a practical application of the standard deviation. It is useful if you wish to know the probability of a large net loss or win after a session of flat betting. The left column represents the number of hands in the session. The top row represents the probability that the result, after adjusting for the house edge, will exceed the table value. The body of the table represents the number of units won or lost, after adjusting for the house edge.
For example suppose a blackjack player loses 100 units over a session of 1000 bets. Assuming an 0.4% house edge, 4 of the losses are expected due to the house edge and 96 are the result of bad luck. The player wishes to know the probability of a loss of this magnitude. The table shows the probability of a loss of 95 units to be 0.5%. Thus the player can expect to lose 95 units or more about 1 session in 200.
<CENTER><TABLE cellSpacing=2 cellPadding=1 bgColor=#e0d18b border=1><TBODY><TR><TD align=middle colSpan=11>Probability of Loss Table</TD></TR><TR><TD align=right>Number
of Hands</TD><TD align=middle>10%</TD><TD align=middle>5%</TD><TD align=middle>2.5%</TD><TD align=middle>1%</TD><TD align=middle>0.5%</TD><TD align=middle>0.25%</TD><TD align=middle>0.1%</TD><TD align=middle>0.05%</TD><TD align=middle>0.01%</TD></TR><TR><TD align=right>100</TD><TD align=right>15</TD><TD align=right>19</TD><TD align=right>23</TD><TD align=right>27</TD><TD align=right>30</TD><TD align=right>33</TD><TD align=right>36</TD><TD align=right>39</TD><TD align=right>43</TD></TR><TR><TD align=right>200</TD><TD align=right>21</TD><TD align=right>27</TD><TD align=right>32</TD><TD align=right>39</TD><TD align=right>43</TD><TD align=right>46</TD><TD align=right>51</TD><TD align=right>55</TD><TD align=right>60</TD></TR><TR><TD align=right>300</TD><TD align=right>26</TD><TD align=right>33</TD><TD align=right>40</TD><TD align=right>47</TD><TD align=right>52</TD><TD align=right>57</TD><TD align=right>63</TD><TD align=right>67</TD><TD align=right>74</TD></TR><TR><TD align=right>400</TD><TD align=right>30</TD><TD align=right>38</TD><TD align=right>46</TD><TD align=right>54</TD><TD align=right>60</TD><TD align=right>66</TD><TD align=right>73</TD><TD align=right>77</TD><TD align=right>85</TD></TR><TR><TD align=right>500</TD><TD align=right>33</TD><TD align=right>43</TD><TD align=right>51</TD><TD align=right>61</TD><TD align=right>67</TD><TD align=right>73</TD><TD align=right>81</TD><TD align=right>86</TD><TD align=right>95</TD></TR><TR><TD align=right>600</TD><TD align=right>37</TD><TD align=right>47</TD><TD align=right>56</TD><TD align=right>67</TD><TD align=right>74</TD><TD align=right>80</TD><TD align=right>89</TD><TD align=right>95</TD><TD align=right>105</TD></TR><TR><TD align=right>700</TD><TD align=right>40</TD><TD align=right>51</TD><TD align=right>61</TD><TD align=right>72</TD><TD align=right>80</TD><TD align=right>87</TD><TD align=right>96</TD><TD align=right>102</TD><TD align=right>113</TD></TR><TR><TD align=right>800</TD><TD align=right>42</TD><TD align=right>54</TD><TD align=right>65</TD><TD align=right>77</TD><TD align=right>85</TD><TD align=right>93</TD><TD align=right>103</TD><TD align=right>109</TD><TD align=right>121</TD></TR><TR><TD align=right>900</TD><TD align=right>45</TD><TD align=right>58</TD><TD align=right>69</TD><TD align=right>82</TD><TD align=right>91</TD><TD align=right>99</TD><TD align=right>109</TD><TD align=right>116</TD><TD align=right>128</TD></TR><TR><TD align=right>1000</TD><TD align=right>47</TD><TD align=right>61</TD><TD align=right>72</TD><TD align=right>86</TD><TD align=right>95</TD><TD align=right>104</TD><TD align=right>115</TD><TD align=right>122</TD><TD align=right>135</TD></TR><TR><TD align=right>2000</TD><TD align=right>67</TD><TD align=right>86</TD><TD align=right>103</TD><TD align=right>122</TD><TD align=right>135</TD><TD align=right>147</TD><TD align=right>162</TD><TD align=right>173</TD><TD align=right>191</TD></TR><TR><TD align=right>3000</TD><TD align=right>82</TD><TD align=right>105</TD><TD align=right>126</TD><TD align=right>149</TD><TD align=right>165</TD><TD align=right>180</TD><TD align=right>199</TD><TD align=right>211</TD><TD align=right>234</TD></TR><TR><TD align=right>4000</TD><TD align=right>95</TD><TD align=right>122</TD><TD align=right>145</TD><TD align=right>172</TD><TD align=right>191</TD><TD align=right>208</TD><TD align=right>229</TD><TD align=right>244</TD><TD align=right>270</TD></TR><TR><TD align=right>5000</TD><TD align=right>106</TD><TD align=right>136</TD><TD align=right>162</TD><TD align=right>193</TD><TD align=right>213</TD><TD align=right>232</TD><TD align=right>256</TD><TD align=right>273</TD><TD align=right>302</TD></TR><TR><TD align=right>6000</TD><TD align=right>116</TD><TD align=right>149</TD><TD align=right>178</TD><TD align=right>211</TD><TD align=right>234</TD><TD align=right>255</TD><TD align=right>281</TD><TD align=right>299</TD><TD align=right>331</TD></TR><TR><TD align=right>7000</TD><TD align=right>125</TD><TD align=right>161</TD><TD align=right>192</TD><TD align=right>228</TD><TD align=right>252</TD><TD align=right>275</TD><TD align=right>303</TD><TD align=right>323</TD><TD align=right>357</TD></TR><TR><TD align=right>8000</TD><TD align=right>134</TD><TD align=right>172</TD><TD align=right>205</TD><TD align=right>244</TD><TD align=right>270</TD><TD align=right>294</TD><TD align=right>324</TD><TD align=right>345</TD><TD align=right>382</TD></TR><TR><TD align=right>9000</TD><TD align=right>142</TD><TD align=right>183</TD><TD align=right>217</TD><TD align=right>259</TD><TD align=right>286</TD><TD align=right>312</TD><TD align=right>344</TD><TD align=right>366</TD><TD align=right>405</TD></TR><TR><TD align=right>10000</TD><TD align=right>150</TD><TD align=right>192</TD><TD align=right>229</TD><TD align=right>272</TD><TD align=right>302</TD><TD align=right>329</TD><TD align=right>363</TD><TD align=right>386</TD><TD align=right>427</TD></TR><TR><TD align=right>20000</TD><TD align=right>212</TD><TD align=right>272</TD><TD align=right>324</TD><TD align=right>385</TD><TD align=right>427</TD><TD align=right>465</TD><TD align=right>513</TD><TD align=right>546</TD><TD align=right>604</TD></TR><TR><TD align=right>30000</TD><TD align=right>259</TD><TD align=right>333</TD><TD align=right>397</TD><TD align=right>472</TD><TD align=right>523</TD><TD align=right>569</TD><TD align=right>628</TD><TD align=right>668</TD><TD align=right>739</TD></TR><TR><TD align=right>40000</TD><TD align=right>299</TD><TD align=right>385</TD><TD align=right>458</TD><TD align=right>545</TD><TD align=right>603</TD><TD align=right>657</TD><TD align=right>725</TD><TD align=right>772</TD><TD align=right>854</TD></TR><TR><TD align=right>50000</TD><TD align=right>335</TD><TD align=right>430</TD><TD align=right>513</TD><TD align=right>609</TD><TD align=right>675</TD><TD align=right>735</TD><TD align=right>811</TD><TD align=right>863</TD><TD align=right>955</TD></TR><TR><TD align=right>60000</TD><TD align=right>367</TD><TD align=right>471</TD><TD align=right>561</TD><TD align=right>667</TD><TD align=right>739</TD><TD align=right>805</TD><TD align=right>888</TD><TD align=right>945</TD><TD align=right>1046</TD></TR><TR><TD align=right>70000</TD><TD align=right>396</TD><TD align=right>509</TD><TD align=right>606</TD><TD align=right>721</TD><TD align=right>798</TD><TD align=right>869</TD><TD align=right>959</TD><TD align=right>1021</TD><TD align=right>1129</TD></TR><TR><TD align=right>80000</TD><TD align=right>423</TD><TD align=right>544</TD><TD align=right>648</TD><TD align=right>771</TD><TD align=right>853</TD><TD align=right>930</TD><TD align=right>1025</TD><TD align=right>1092</TD><TD align=right>1207</TD></TR><TR><TD align=right>90000</TD><TD align=right>449</TD><TD align=right>577</TD><TD align=right>688</TD><TD align=right>817</TD><TD align=right>905</TD><TD align=right>986</TD><TD align=right>1088</TD><TD align=right>1158</TD><TD align=right>1281</TD></TR><TR><TD align=right>100000</TD><TD align=right>473</TD><TD align=right>608</TD><TD align=right>725</TD><TD align=right>862</TD><TD align=right>954</TD><TD align=right>1039</TD><TD align=right>1146</TD><TD align=right>1220</TD><TD align=right>1350</TD></TR><TR><TD align=right>200000</TD><TD align=right>669</TD><TD align=right>860</TD><TD align=right>1025</TD><TD align=right>1219</TD><TD align=right>1349</TD><TD align=right>1470</TD><TD align=right>1621</TD><TD align=right>1726</TD><TD align=right>1909</TD></TR><TR><TD align=right>300000</TD><TD align=right>820</TD><TD align=right>1054</TD><TD align=right>1256</TD><TD align=right>1493</TD><TD align=right>1653</TD><TD align=right>1800</TD><TD align=right>1986</TD><TD align=right>2114</TD><TD align=right>2338</TD></TR><TR><TD align=right>400000</TD><TD align=right>947</TD><TD align=right>1217</TD><TD align=right>1450</TD><TD align=right>1723</TD><TD align=right>1908</TD><TD align=right>2078</TD><TD align=right>2293</TD><TD align=right>2441</TD><TD align=right>2700</TD></TR><TR><TD align=right>500000</TD><TD align=right>1059</TD><TD align=right>1360</TD><TD align=right>1621</TD><TD align=right>1927</TD><TD align=right>2134</TD><TD align=right>2324</TD><TD align=right>2564</TD><TD align=right>2729</TD><TD align=right>3018</TD></TR><TR><TD align=right>600000</TD><TD align=right>1160</TD><TD align=right>1490</TD><TD align=right>1776</TD><TD align=right>2111</TD><TD align=right>2337</TD><TD align=right>2546</TD><TD align=right>2808</TD><TD align=right>2990</TD><TD align=right>3307</TD></TR><TR><TD align=right>700000</TD><TD align=right>1252</TD><TD align=right>1610</TD><TD align=right>1918</TD><TD align=right>2280</TD><TD align=right>2525</TD><TD align=right>2750</TD><TD align=right>3033</TD><TD align=right>3229</TD><TD align=right>3572</TD></TR><TR><TD align=right>800000</TD><TD align=right>1339</TD><TD align=right>1721</TD><TD align=right>2050</TD><TD align=right>2437</TD><TD align=right>2699</TD><TD align=right>2939</TD><TD align=right>3243</TD><TD align=right>3452</TD><TD align=right>3818</TD></TR><TR><TD align=right>900000</TD><TD align=right>1420</TD><TD align=right>1825</TD><TD align=right>2175</TD><TD align=right>2585</TD><TD align=right>2863</TD><TD align=right>3118</TD><TD align=right>3439</TD><TD align=right>3661</TD><TD align=right>4050</TD></TR><TR><TD align=right>1000000</TD><TD align=right>1497</TD><TD align=right>1924</TD><TD align=right>2292</TD><TD align=right>2725</TD><TD align=right>3017</TD><TD align=right>3286</TD><TD align=right>3626</TD><TD align=right>3859</TD><TD align=right>4269</TD></TR></TBODY></TABLE></CENTER>