# Ask The Wizard #331

Click the button below for the answer.

Here is my solution (PDF).

This problem was asked and discussed in my forum at Wizard of Vegas.

It was adapted from the puzzle, Can You Join the World’s Biggest Zoom Call? at FiveThirtyEight.

It is extremely unlikely to find a blackjack game that allows drawing to split aces, to be dealt a pair of aces and then reach the splitting limit. Nevertheless, I endeavor to address the most obscure situations and admit my basic strategy tables at the time of this question didn’t address what to do in this situation.

The answer is to hit, except double if:

- The dealer has a six up (with any number of decks)
- The dealer has a five up with one or two decks.

Here is the expected value of this situation under various such situations.

### Expected Value of Hitting and Doubling Soft 12

Decks | Stand Soft 17 |
Dealer Up Card |
Hit EV |
Double EV |
Best Play |
---|---|---|---|---|---|

1 | Stand | 5 | 0.182014 | 0.215727 | Double |

1 | Hit | 5 | 0.182058 | 0.215933 | Double |

1 | Stand | 6 | 0.199607 | 0.247914 | Double |

1 | Hit | 6 | 0.201887 | 0.258415 | Double |

2 | Stand | 5 | 0.169241 | 0.170637 | Double |

2 | Hit | 5 | 0.169339 | 0.171311 | Double |

2 | Stand | 6 | 0.192311 | 0.213109 | Double |

2 | Hit | 6 | 0.194397 | 0.227011 | Double |

4 | Stand | 5 | 0.162849 | 0.148228 | Hit |

4 | Hit | 5 | 0.162955 | 0.149183 | Hit |

4 | Stand | 6 | 0.18902 | 0.196249 | Double |

4 | Hit | 6 | 0.19074 | 0.211466 | Double |

Expected values taken from my blackjack hand calculator.

The answer is a candidate can receive as little as 21.69% of the popular vote and still win.

To elaborate, the following table shows the state by state population and electoral votes by state. The population numbers are taken as of 2019 and the electoral votes the last time they were adjusted in 2010. As a reminder to my readers outside of the United States, each state also gets a bonus two electoral votes. The result is states with a small population have much more influence on elections than those with a large population. As of the 2020 election, voters in Wyoming have almost four times the influence in the presidential election as those in Texas.

Given the rules, a candidate could get 100% of the vote in Texas, Florida, California ,North Carolina, New York, Georgia, Arizona, Virginia, Ohio, Pennsylvania, New Jersey, and Missouri, plus get half the vote (less one) in every other state and secure a total of 257,085,170 popular votes. Meanwhile, the other candidate would get 71,215,374 only, and win with exactly the needed 270 electoral votes.

The following table breaks it down. It is listed in order of million population per electoral vote (least to greatest).

### Electoral College Hypothetical Scenario

State | Population | Electoral Votes |
Million People per Electoral Vote |
Votes for A | Votes for B |
---|---|---|---|---|---|

Texas | 28,995,881 | 38 | 1.311 | - | 28,995,881 |

Florida | 21,477,737 | 29 | 1.350 | - | 21,477,737 |

California | 39,512,223 | 55 | 1.392 | - | 39,512,223 |

North Carolina | 10,488,084 | 15 | 1.430 | - | 10,488,084 |

New York | 19,453,561 | 29 | 1.491 | - | 19,453,561 |

Georgia | 10,617,423 | 16 | 1.507 | - | 10,617,423 |

Arizona | 7,278,717 | 11 | 1.511 | - | 7,278,717 |

Virginia | 8,535,519 | 13 | 1.523 | - | 8,535,519 |

Ohio | 11,689,100 | 18 | 1.540 | - | 11,689,100 |

Pennsylvania | 12,801,989 | 20 | 1.562 | - | 12,801,989 |

Colorado | 5,758,736 | 9 | 1.563 | 2,879,369 | 2,879,367 |

Washington | 7,614,893 | 12 | 1.576 | 3,807,447 | 3,807,446 |

New Jersey | 8,882,190 | 14 | 1.576 | - | 8,882,190 |

Illinois | 12,671,821 | 20 | 1.578 | 6,335,911 | 6,335,910 |

Massachusetts | 6,949,503 | 11 | 1.583 | 3,474,752 | 3,474,751 |

Michigan | 9,986,857 | 16 | 1.602 | 4,993,429 | 4,993,428 |

Tennessee | 6,833,174 | 11 | 1.610 | 3,416,588 | 3,416,586 |

Missouri | 6,137,428 | 10 | 1.629 | - | 6,137,428 |

Indiana | 6,732,219 | 11 | 1.634 | 3,366,110 | 3,366,109 |

Maryland | 6,045,680 | 10 | 1.654 | 3,022,841 | 3,022,839 |

Oregon | 4,217,737 | 7 | 1.660 | 2,108,869 | 2,108,868 |

Wisconsin | 5,822,434 | 10 | 1.717 | 2,911,218 | 2,911,216 |

Louisiana | 4,648,794 | 8 | 1.721 | 2,324,398 | 2,324,396 |

South Carolina | 5,148,714 | 9 | 1.748 | 2,574,358 | 2,574,356 |

Oklahoma | 3,956,971 | 7 | 1.769 | 1,978,486 | 1,978,485 |

Minnesota | 5,639,632 | 10 | 1.773 | 2,819,817 | 2,819,815 |

Kentucky | 4,467,673 | 8 | 1.791 | 2,233,837 | 2,233,836 |

Alabama | 4,903,185 | 9 | 1.836 | 2,451,593 | 2,451,592 |

Utah | 3,205,958 | 6 | 1.872 | 1,602,980 | 1,602,978 |

Iowa | 3,155,070 | 6 | 1.902 | 1,577,536 | 1,577,534 |

Nevada | 3,080,156 | 6 | 1.948 | 1,540,079 | 1,540,077 |

Connecticut | 3,565,287 | 7 | 1.963 | 1,782,644 | 1,782,643 |

Arkansas | 3,017,825 | 6 | 1.988 | 1,508,913 | 1,508,912 |

Mississippi | 2,976,149 | 6 | 2.016 | 1,488,075 | 1,488,074 |

Kansas | 2,913,314 | 6 | 2.060 | 1,456,658 | 1,456,656 |

Idaho | 1,787,065 | 4 | 2.238 | 893,533 | 893,532 |

New Mexico | 2,096,829 | 5 | 2.385 | 1,048,415 | 1,048,414 |

Nebraska | 1,934,408 | 5 | 2.585 | 967,205 | 967,203 |

West Virginia | 1,792,147 | 5 | 2.790 | 896,074 | 896,073 |

Montana | 1,068,778 | 3 | 2.807 | 534,390 | 534,388 |

Hawaii | 1,415,872 | 4 | 2.825 | 707,937 | 707,935 |

New Hampshire | 1,359,711 | 4 | 2.942 | 679,856 | 679,855 |

Maine | 1,344,212 | 4 | 2.976 | 672,107 | 672,105 |

Delaware | 973,764 | 3 | 3.081 | 486,883 | 486,881 |

South Dakota | 884,659 | 3 | 3.391 | 442,330 | 442,329 |

Rhode Island | 1,059,361 | 4 | 3.776 | 529,681 | 529,680 |

North Dakota | 762,062 | 3 | 3.937 | 381,032 | 381,030 |

Alaska | 731,545 | 3 | 4.101 | 365,773 | 365,772 |

DC | 705,749 | 3 | 4.251 | 352,875 | 352,874 |

Vermont | 623,989 | 3 | 4.808 | 311,995 | 311,994 |

Wyoming | 578,759 | 3 | 5.184 | 289,380 | 289,379 |

Total | 328,300,544 | 538 | 71,215,374 | 257,085,170 |

Sources:

The answer is 219.149467.

There are two ways I can think of to solve this. The first is with a Markov Chain. The following table shows the expected rolls needed from any given state of the 128 possible.

### Fire Bet — Markov Chain

Point 4 Made |
Point 5 Made |
Point 6 Made |
Point 8 Made |
Point 9 Made |
Point 10 Made |
Expected Rolls |
---|---|---|---|---|---|---|

No | No | No | No | No | No | 219.149467 |

No | No | No | No | No | Yes | 183.610129 |

No | No | No | No | Yes | No | 208.636285 |

No | No | No | No | Yes | Yes | 168.484195 |

No | No | No | Yes | No | No | 215.452057 |

No | No | No | Yes | No | Yes | 177.801038 |

No | No | No | Yes | Yes | No | 203.975216 |

No | No | No | Yes | Yes | Yes | 160.639243 |

No | No | Yes | No | No | No | 215.452057 |

No | No | Yes | No | No | Yes | 177.801038 |

No | No | Yes | No | Yes | No | 203.975216 |

No | No | Yes | No | Yes | Yes | 160.639243 |

No | No | Yes | Yes | No | No | 211.272344 |

No | No | Yes | Yes | No | Yes | 170.911638 |

No | No | Yes | Yes | Yes | No | 198.520513 |

No | No | Yes | Yes | Yes | Yes | 150.740559 |

No | Yes | No | No | No | No | 208.636285 |

No | Yes | No | No | No | Yes | 168.484195 |

No | Yes | No | No | Yes | No | 196.113524 |

No | Yes | No | No | Yes | Yes | 149.383360 |

No | Yes | No | Yes | No | No | 203.975216 |

No | Yes | No | Yes | No | Yes | 160.639243 |

No | Yes | No | Yes | Yes | No | 189.938796 |

No | Yes | No | Yes | Yes | Yes | 137.865939 |

No | Yes | Yes | No | No | No | 203.975216 |

No | Yes | Yes | No | No | Yes | 160.639243 |

No | Yes | Yes | No | Yes | No | 189.938796 |

No | Yes | Yes | No | Yes | Yes | 137.865939 |

No | Yes | Yes | Yes | No | No | 198.520513 |

No | Yes | Yes | Yes | No | Yes | 150.740559 |

No | Yes | Yes | Yes | Yes | No | 182.290909 |

No | Yes | Yes | Yes | Yes | Yes | 121.527273 |

Yes | No | No | No | No | No | 183.610129 |

Yes | No | No | No | No | Yes | 136.890807 |

Yes | No | No | No | Yes | No | 168.484195 |

Yes | No | No | No | Yes | Yes | 113.177130 |

Yes | No | No | Yes | No | No | 177.801038 |

Yes | No | No | Yes | No | Yes | 126.849235 |

Yes | No | No | Yes | Yes | No | 160.639243 |

Yes | No | No | Yes | Yes | Yes | 98.046264 |

Yes | No | Yes | No | No | No | 177.801038 |

Yes | No | Yes | No | No | Yes | 126.849235 |

Yes | No | Yes | No | Yes | No | 160.639243 |

Yes | No | Yes | No | Yes | Yes | 98.046264 |

Yes | No | Yes | Yes | No | No | 170.911638 |

Yes | No | Yes | Yes | No | Yes | 113.931818 |

Yes | No | Yes | Yes | Yes | No | 150.740559 |

Yes | No | Yes | Yes | Yes | Yes | 75.954545 |

Yes | Yes | No | No | No | No | 168.484195 |

Yes | Yes | No | No | No | Yes | 113.177130 |

Yes | Yes | No | No | Yes | No | 149.383360 |

Yes | Yes | No | No | Yes | Yes | 80.208000 |

Yes | Yes | No | Yes | No | No | 160.639243 |

Yes | Yes | No | Yes | No | Yes | 98.046264 |

Yes | Yes | No | Yes | Yes | No | 137.865939 |

Yes | Yes | No | Yes | Yes | Yes | 53.472000 |

Yes | Yes | Yes | No | No | No | 160.639243 |

Yes | Yes | Yes | No | No | Yes | 98.046264 |

Yes | Yes | Yes | No | Yes | No | 137.865939 |

Yes | Yes | Yes | No | Yes | Yes | 53.472000 |

Yes | Yes | Yes | Yes | No | No | 150.740559 |

Yes | Yes | Yes | Yes | No | Yes | 75.954545 |

Yes | Yes | Yes | Yes | Yes | No | 121.527273 |

Yes | Yes | Yes | Yes | Yes | Yes | 0.000000 |

Briefly, the expected rolls from any given state is the expected rolls until point is either made or lost (5.063636) plus the expected number of rolls if the player advances to a further state, divided by the probability of not advancing in state.

The other method uses integral calculus. First calculate the expected rolls for each possible outcome to happen. Then take the dot product of the probability of each event and average rolls to get the average rolls to resolve a pass line bet, which the lower right corner shows is 3.375758 = 557/165.

### Fire Bet — Expected Rolls

Event | Probability | Average Rolls | Expected Rolls |
---|---|---|---|

Point 4 win | 0.027778 | 5 | 0.138889 |

pt 5 win | 0.044444 | 4.6 | 0.204444 |

pt 6 win | 0.063131 | 4.272727 | 0.269743 |

pt 8 win | 0.063131 | 4.272727 | 0.269743 |

pt 9 win | 0.044444 | 4.6 | 0.204444 |

pt 10 win | 0.027778 | 5 | 0.138889 |

pt 4 loss | 0.055556 | 5 | 0.277778 |

pt 5 loss | 0.066667 | 4.6 | 0.306667 |

pt 6 loss | 0.075758 | 4.272727273 | 0.323691 |

pt 8 loss | 0.075758 | 4.272727273 | 0.323691 |

pt 9 loss | 0.066667 | 4.6 | 0.306667 |

pt 10 loss | 0.055556 | 5 | 0.277778 |

Come out roll win | 0.222222 | 1 | 0.222222 |

Come out roll loss | 0.111111 | 1 | 0.111111 |

Total | 1.000000 | 3.375758 |

From there we can get the expected rolls between any given point winning:

- Rolls between a point of 4 winning = (3/36)*(3/9)*5*(557/165) = 6684/55 = apx 121.527273.
- Rolls between a point of 5 winning = (4/36)*(4/10)*4.6*(557/165) = 1671/21 = apx 75.954545.
- Rolls between a point of 6 winning = (5/36)*(5/11)*(47/11)*(557/165) = 6684/125 = apx 53.472.

The expected rolls for a 10, 9, and 8 point winner are the same as for 4, 5, and 6, respectively.

Let's say that instead of a point-4 winner happening on a discrete basis, it follows an exponential distribution with a mean of 6684/55. The probability such a random variable lasts x units of time without happening is exp(-x/(6684/55)) = exp(-55x/6684).

The probability it has happened within x units of time, at least once, is 1-exp(-55x/6684).

If we represent all six points as continuous variables, then the probability all six have happened within x units of time is (1-exp(-55x/6684))^2 * (1-exp(-22x/1671))^2 * (1-exp(-125x/6684))^2.

The probability at least one of the six events not happening within x units of time is 1 - (1-exp(-55x/6684))^2 * (1-exp(-22x/1671))^2 * (1-exp(-125x/6684))^2.

We can get the expected time for all six events to happen by integrating the above from 0 to infinity.

Using this integral calculator gives an answer of 8706865474775503638338329687/39730260732259873692189000 = apx 219.1494672902.

Why this works is harder to explain, so please take that part on faith.