Ask The Wizard #303
Did you know that Dream Card video poker games do not give the correct Dream Card in Bonus Poker when the four dealt cards are 5 singletons, with the lowest a four and the other three all less than a jack, with no straight or flush chance? The correct Dream Card would be the four, because of the premium win on a four fours. However, the Dream Card given will be one of the other ranks. What is the cost of this incorrect Dream Card, assuming the player always accepts it?
Yes, I was aware of this. For the benefit of the other readers, in Dream Card the game will sometimes give the optimal 5th card on the deal based on the first four randomly dealt cards. The probability of getting a Dream Card in Bonus Poker is 46.7%. The player can always reject the suggested Dream Card and switch to any other card still in the deck. However, the advice is always correct, as far as I know, except in this situation with Bonus Poker.
If the player gets the Dream Card, then the probability the other four cards will fall into this situation is 1.49%. Given the 46.7% probability of the Dream Card, this situation will happen with probability 0.70% or once every 144 hands.
Using my video poker hand analyzer, the expected value of a pair of fours in 8/5 Bonus Poker is 0.855134. The expected value of a pair of 5's to 10's is 0.813506. So the cost of the bug every time it occurs, assuming the player accepts the Dream Card, is 0.041628 in expected value.
The overall cost to the return of the game is 0.006955 × 0.041628 = 0.000290, or about 0.03%.
If the player gets the Dream Card, then the probability the other four cards will fall into this situation is 1.49%. Given the 46.7% probability of the Dream Card, this situation will happen with probability 0.70% or once every 144 hands.
Using my video poker hand analyzer, the expected value of a pair of fours in 8/5 Bonus Poker is 0.855134. The expected value of a pair of 5's to 10's is 0.813506. So the cost of the bug every time it occurs, assuming the player accepts the Dream Card, is 0.041628 in expected value.
The overall cost to the return of the game is 0.006955 × 0.041628 = 0.000290, or about 0.03%.
Which starting hand has the highest chance of winning against a pair of random aces in a two-player game of Texas Hold 'Em?
Using my Texas Hold 'Em calculator, I show the answer is suited 5-6. If the 5-6 is in a different suit from both aces, the probability of winning is 22.87% and a tie 00.37%. If the 5-6 is in the same suit as one of the aces, then the probability of winning is 21.71% and a tie 00.46%. On average, the player with 5-6 suited will lose 0.55005 units, assuming he bet one unit, and have a chance of winning, given that there is a winner, of 22.383%.
In the Hot Roll bonus, the player wins the following number of coins according to the total of two dice:
He keeps rolling until he gets a total of seven, which ends the bonus. If he rolls a seven on the first roll, then he gets a consolation prize of 700 coins. What is the average coins won per bonus?
- 2 or 12: 1000
- 3 or 11: 600
- 4 or 10: 400
- 5 or 9: 300
- 6 or 8: 200
He keeps rolling until he gets a total of seven, which ends the bonus. If he rolls a seven on the first roll, then he gets a consolation prize of 700 coins. What is the average coins won per bonus?
The average number of rolls is the inverse of the bonus-ending event, which has a probability of 1/6, so the player will roll six times on average. However, the last roll will be the seven, so an average of five winning rolls per bonus.
Next, here is the probability of each total, assuming no seven:
So, the average win per roll, assuming no seven, is 2*[(1/30)*1000 + (2/30)*600 + (3/30)*400 + (4/30)*300 + (5/30)*200] = 373.33.
The value of the consolation prize is (1/6)*700 = 116.67.
Thus, the average bonus win is 116.67 + 5×373.33 = 1983.33.
Next, here is the probability of each total, assuming no seven:
- 2 or 12: 1/30
- 3 or 11: 2/30
- 4 or 10: 3/30
- 5 or 9: 4/30
- 6 or 8: 5/30
So, the average win per roll, assuming no seven, is 2*[(1/30)*1000 + (2/30)*600 + (3/30)*400 + (4/30)*300 + (5/30)*200] = 373.33.
The value of the consolation prize is (1/6)*700 = 116.67.
Thus, the average bonus win is 116.67 + 5×373.33 = 1983.33.