Ask The Wizard #299

Is there any hand in video poker where there is a tie for the highest expected value but not a tie in variance?


Yes! There are lots of situations where there is a tie for the highest expected value. For example, a dealt four of a kind in Jacks or Better. It doesn't make any difference whether you hold the kicker or not. Another is with a dealt two pair in full pay deuces wild. The correct play is to hold just one of the pairs, and it doesn't matter which one. However, in both these examples the chances of each possible outcome is the same on the draw.

A hand where there is a difference in variance is in full pay deuces wild with a hand that could be played as three to a straight flush with two gaps or four to an inside straight. For example, suited 8-6-4 with an off-suit 7 and king. The following two tables show the expected return of each viable play.

Holding Three to a Straight Flush

Hand Pays Combinations Probability Return
Straight flush 9 15 0.013876 0.124884
Flush 2 63 0.058279 0.116559
Straight 2 31 0.028677 0.057354
Three of a kind 1 45 0.041628 0.041628
Loss 0 927 0.857539 0.000000
Total 1081 1.000000 0.340426

Holding Four to a Straight

Hand Pays Combinations Probability Return
Straight 2 8 0.170213 0.340426
Loss 0 39 0.829787 0.000000
Total 47 1.000000 0.340426

The bottom right cell of each table shows an expected return of 16/47 (34.04%) for each hand. However, the variance of holding four to the straight is 0.564962 and three to the straight flush is 1.397524.

My thanks to Bob Dancer for bringing this hand to my attention.

What is the mean distance between two random points in a unit square?


For a question so easy to ask, the solution is rather involved. The way I did it, you will need to know this integral.

Here is the answer and my solution (PDF).

What is the cost to not playing all four lines in Multi-Strike poker?

Rob F. from Las Vegas

Let's look at 8-5 Bonus Poker as an example. The following table shows the return by number of lines bet.

  • 4 lines: 99.375%
  • 3 lines: 99.279%
  • 2 lines: 99.214%
  • 1 line: 99.166%

The next list shows the cost to not playing the maximum lines according to how many lines are played.

  • 4 lines: 0.000%
  • 3 lines: 0.095%
  • 2 lines: 0.160%
  • 1 line: 0.209%

Harrah's in Philadelphia is paying the following bonuses in blackjack:

Harrah's Philadelphia Promotion

Hand Pays
Triple sevens $500
Five-card 21 $250
Black ace and black jack $150
Red ace and black jack $100
Suited blackjack $50

To get the bonuses a minimum bet of $25 is required. Six decks are used. Can you tell me the value of this promotion?


Nice promotion! The following table shows the probability of each event. The probability for the five-card 21 should be considered a bit rough.

Harrah's Philadelphia Promotion Analysis

Hand Pays Probability Return
Triple sevens $500 0.000384552 $0.19
Five-card 21 $250 0.00453345 $1.13
Black ace and black jack $150 0.002968093 $0.45
Red ace and black jack $100 0.002968093 $0.30
Suited blackjack $50 0.011872372 $0.59
Total $- 0.011872372 $2.66

The bottom right cell shows the bonuses are worth $2.66 per hand played.

The blackjack rules are pretty liberal there, with a house edge of only 0.35%. On the minimum required bet of $25, the expected loss per hand is $0.08. So, the promotion is worth $2.57 per hand played.

Unfortunately, the promotion has ended as of this publication date.

This question was raised and discussed in my forum at Wizard of Vegas.