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Ask The Wizard #299
Are there situations in video poker where two hands yield the same highest expected value but differ in terms of variance?
Absolutely! There are numerous instances where hands can tie in their expected value. A typical example is being dealt four of a kind in Jacks or Better; it doesn’t change your outcome whether you decide to keep the kicker or not. Another scenario involves receiving two pairs in full pay deuces wild, where the right decision is to hold one of the pairs, regardless of which one. In both cases, the probability of each possible draw outcome remains consistent.
An example highlighting a variance difference arises in full pay deuces wild, with a hand that could form either three to a straight flush with two gaps or four to an inside straight. Consider holding suited cards like 8-6-4 alongside an off-suit 7 and king. The subsequent tables will detail the expected returns for each feasible action.
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 in both tables indicates an expected return of 16/47 (34.04%) for each hand option. Yet, the variance for holding four to the straight is calculated at 0.564962, while for three to the straight flush, it stands at 1.397524.
I appreciate Bob Dancer for highlighting this particular hand.
What is the average distance between two randomly selected points within a unit square?
Although this question appears straightforward, the answer is quite complex. The method I employed requires understanding. this integral .
Here is the answer and my solution (PDF) .
What are the implications of not wagering on all four lines in this scenario? Multi-Strike poker ?
Let's use 8-5 Bonus Poker as an illustration. The following chart demonstrates the returns relative to the number of lines wagered.
- 4 lines: 99.375%
- 3 lines: 99.279%
- 2 lines: 99.214%
- 1 line: 99.166%
The subsequent list outlines the financial repercussions of not playing the maximum number of lines based on the lines actually played.
- 4 lines: 0.000%
- 3 lines: 0.095%
- 2 lines: 0.160%
- 1 line: 0.209%
Harrah's in Philadelphia offers various blackjack bonuses.
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 qualify for these bonuses, a minimum bet of $25 is necessary, and the game is played with six decks. Can you assess the value of this offer?
What an enticing promotion! The table below presents the likelihood of each outcome, with the probability of achieving a five-card 21 being somewhat approximate.
Analysis of Harrah's Philadelphia Promotion
| 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 indicates that each bonus is valued at $2.66 for every hand dealt.
The blackjack rules at this venue are quite favorable, featuring a house edge of merely 0.35%. With the required minimum bet of $25, the expected loss per hand amounts to $0.08, making the promotion worth approximately $2.57 for each hand played.
Regrettably, this promotion came to a close as of the date of this publication.
This inquiry sparked a discussion within my forum at Wizard of Vegas .