Michael Shackleford, who is also known as the 'Wizard of Odds' and the author of 'Gambling 102,' gives a quick rundown on the best strategies for playing fou. The Wizard of Odds answers readers questions. Sign Up For Updates You're Subscribed! Online casino payment. Enter your email address below to subscribe to our weekly newsletter along with other special announcements from The Wizard of Odds!
I read about the Reversible Royal game with the 105.22% return in your article at Wizard of Vegas. That return assumes optimal strategy, including for card order. What is the return if I assuming an average royal win? How about if I use ordinary 6-5 Bonus Poker strategy, which is the base pay table?
Assuming no strategy deviations, 1 in 60 royals will be sequential. The reversible royal jackpot pays 161,556 for 1. Any other royal pays 800 for 1. The average royal win is thus (1/60)*161,556 + (59/60)*800 + 17,396 for 1.
If we assuming all royals pay 17,396 and play optimal strategy based on that royal win, then the return drops to 103.56%.
If we play standard 6-5 Bonus Poker strategy, which is the base pay table, then the return drops further to 101.97%.
Consider the unit square with coordinates (0,0), (1,0), (1,1), (0,1). Line A goes from (0,0) to (1,1). Line B goes from (1,0) to 0.5,1). What is the radius of the circle tangent to lines A, B, and the bottom of the circle?
![Fun Fun](https://image1-srjcooldude.netdna-ssl.com/wp-content/uploads/2017/11/craps-game-winning-odds.jpg)
This puzzle appeared in the October 2020 edition of the Mensa Bulletin.
The answer is (1+sqrt(2)+sqrt(5)+sqrt(50))/12 = apx. 0.248000646617418.
Here is my solution (PDF).
This problem is asked and discussed in my forum at Wizard of Vegas.
What is the probability of getting a Yahtzee if that is the only category you have left on the card?
For the benefit of the readers not familiar with Yahtzee, the question is asking what is the probability of getting a five of a kind in three rolls of five dice. After each roll, you must choose which dice to hold onto and which dice to re-roll.
Here are the possible outcomes after the first roll or any roll where the player rolls 4 or 5 dice.
- Five of a kind = 6*(1/6)^5 = 0.000772
- Four of a kind = (1/6)^3*(5/6)*4 = 0.015432
- Three of a kind = (1/6)^2*(5/6)^2*COMBIN(4,2) = 0.115741
- Two of a kind = 4*(1/6)*(5/6)^3 = 0.385802
- One of a kind = 6*5!/6^5 = 0.092593
Here are the probabilities after holding a pair.
- Five of a kind =(1/6)^3 = 0.004630
- Four of a kind = 3*(1/6)^2*(5/6) = 0.069444
- Three of a kind = 3*(1/6)*(5/6)^2+5*(1/6)^3 = 0.370370
- Two of a kind = (5/6)^3-5*(1/6)^3 = 0.555555
Here are the probabilities after holding a three of a kind:
- Five of a kind =(1/6)^3 = 0.002778
- Four of a kind = 2*(1/6)*(5/6) = 0.27778
- Three of a kind = (5/6)^2 = 0.694444
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Here are the probabilities after holding a four of a kind: Beat the keeper.
- Five of a kind =1/6 = 0.166667
- Four of a kind = 5/6 = 0.83333
With those probabilities of advancement, here are the probabilities of each state after the second roll:
- Five of a kind = 0.000772 + 0.015432*0.166667 + 0.115741*0.002778 + 0.385802*0.004630 + 0.092593* 0.000772 = 0.012631
- Four of a kind = 0.015432*0.166667 + 0.115741*0.27778 + 0.115741*0.27778 = 0.116970
- Three of a kind = 0.115741*0.694444 + 0.385802*0.370370 + 0.092593*0.115741 = 0.409022
- Two of a kind = 0.385802*0.555555 + 0.092593*0.385802 = 0.450103
- One of a kind = 0.092593 * 0.092593 = 0.008573
Using the same probabilities of advancement, here is the probability of a Yahtzee after the third roll:
Five of a kind = 0.012631 + 0.116970*(1/6) + 0.409022*(1/6)^2 + 0.450103*(1/6)^3 + 0.008573*(1/6)^4 = 0.046029. Best casino in vegas to win 2017.
For those of you who prefer matrix algebra, there is the transition matrix:
0.092593 | 0.694444 | 0.192901 | 0.019290 | 0.000772 |
0.000000 | 0.555556 | 0.370370 | 0.069444 | 0.004630 |
0.000000 | 0.000000 | 0.694444 | 0.277778 | 0.027778 |
0.000000 | 0.000000 | 0.000000 | 0.833333 | 0.166667 |
0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 |
If the matrix above is M, then the state after three rolls will be M3, as follows:
0.000794 | 0.256011 | 0.452402 | 0.244765 | 0.046029 |
0.000000 | 0.171468 | 0.435814 | 0.316144 | 0.076575 |
0.000000 | 0.000000 | 0.334898 | 0.487611 | 0.177491 |
0.000000 | 0.000000 | 0.000000 | 0.578704 | 0.421296 |
0.000000 | 0.000000 | 0.000000 | 0.000000 | 1.000000 |
The probability of having a Yahtzee after three rolls can be found in the cell in the upper right corner.
After watching through The Queen's Gambit, I noticed none of the games on the show ended in a draw. I thought chess at high levels had lots of draws. For grandmaster-level chess, what percentage of games end in a draw?
According to the article Has the number of draws in chess increased? at ChessBase.com, author Qiyu Zhou states that in 78,468 rated games between players rated games of 2600 or over (it takes 2500 to be a grandmaster), the following are the results:
- Black wins: 18.0%
- White wins: 28.9%
- Draw: 53.1%
Since craps is a game of chance, you need to understand why you have a greater or lesser chance of rolling different numbers. Because you're rolling two dice, your chances of rolling a specific number in craps are determined by the number of die combinations that can add up to that number. For example, 2 can only be rolled with two 1s, but 4 can be rolled with either a 1 and a 3 or two 2s. That means you have twice the chance of rolling a 3 as you do a 2. Because the 7 has the greatest number of combinations (six), it is the number that has the potential to come up most often, which is why 7 is the magic number in craps.
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There are 36 possible number combinations in craps. Here is a chart showing the possible combinations for each number using two die.
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From the chart, you can see that the most likely number you'll roll is a 7, followed by the 6 and the 8, then 5 and 9, then 4 and 10, then 3 and 11, and finally (and least likely) the 2 and the 12. This means you'll roll a 7 once out of every six rolls, a 6 or an 8 once out of every 7 to 8 rolls, and so on.
Odds for Each Number
By looking at the possible combinations, the 'true odds' for each number can be established. Knowing the odds in craps is good so you have a feel for the likelihood of one number being rolled before another one (e.g., is the 4 going to be rolled before the 7?).
House Edge
Now, true odds are not what the casino pays you unless you're also betting 'free odds' on top of your main bet. Free odds, which is an additional wager you place with your original line bet, pay true odds so the casino's edge is reduced. (We'll talk more about free odds in Strategies the Winners Use.)
To better explain how the casino edge works, let's take the example of flipping a coin. You have a 50/50 chance of the coin landing on heads, and a 50/50 chance of it landing on tails. If that were a bet on which you were being paid true odds, you would be paid even money. The casino, however, has to have an edge in order to make a profit on the game. So, the payoff for any given bet is less than what true mathematical odds would dictate. For example, on a bet that had true odds of 1:1, you would think that if you bet $1 and win, you would be paid $1 in winnings. But in a casino, depending on the bet, you might only be paid $.96. The difference between the true odds and what they pay you is how they make money -- it's called casino odds.
Another way to better understand casino odds versus true odds is to look at the definition of the casino (or house) edge. WizardOfOdds.com defines it as, 'The ratio of the average loss to the initial bet,' going on to explain that it's based on the original wager rather than the average wager so that players can have an idea of how much they are going to lose when they place a bet. For example, by knowing that the casino has a 1.41 percent edge in craps, you can know that you'll be losing 14.1 cents for every $10 bet.
For charts of odds for all types of bets, visit the Wizard of Odds.com.
On the next page, we'll talk about types of craps bets, their odds and what the casino pays for each.