Ask The Wizard #381
It's ironic you test online craps games for fairness, when your own craps game is so seriously flawed. I lost the pass bet by 72 7-out's before I made a point!
To test your theory, I played through 50 pass line bets where the result was either winning by making a point or a 7-out. This took 248 rolls.
In the following table, I document my results.
Craps Data
| Point | Wins | Losses | Total | Prob. Win | Exp. Wins |
|---|---|---|---|---|---|
| 4 | 3 | 5 | 8 | 33.33% | 2.666667 |
| 5 | 3 | 3 | 6 | 40.00% | 2.400000 |
| 6 | 5 | 5 | 10 | 45.45% | 4.545455 |
| 8 | 6 | 4 | 10 | 45.45% | 4.545455 |
| 9 | 4 | 3 | 7 | 40.00% | 2.800000 |
| 10 | 1 | 8 | 9 | 33.33% | 3.000000 |
| Total | 22 | 28 | 50 | 19.957576 |
The right cell shows an expected number of wins of 19.96, given the points rolled. My actual wins was 22. The probability of exactly 22 wins is 7.66%. The probability of less than 22 is 64.73%. The probability of 23 or more is 27.61%. So, this test shows my results were close to expectations.
For all the details, please see this video response to your challenge.
Your basic strategy for six decks and dealer stands on a soft 17 stays to hit a 11 vs. ace. Would would be the right play if the player were limited to just one card if he chooses to hit?
My blackjack hand calculator shows that with six decks and dealer stands on soft 17, player 6+5 vs. dealer ace, the following expected values of each play:
- Hit = +0.147596
- Double = +0.129710
- Stand = -0.661883
Hitting, being limited to one card, would be like doubling, but without doubling the bet. The expected value would thus be half that of doubling, or +0.129710/2 = +0.064855. That is less than the expected value of doubling of +0.129710. Therefore, if you're limited to one card after hitting, then you should double.
You have a 5' ladder. You wish to lean it against a wall as high as possible. However, there is a 1x1x1 cubic foot box you must put the ladder over. No, you can't move the box. What is the maximum height the top of the ladder can reach?
Here is my solution (PDF).