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Ask the Wizard #304

What is the probability of a royal flush if the player always drew to one? Assume if the player always chooses the suit with the most cards already to a royal.

Anonymous

The following table shows the probability of having 0 to 5 cards to a royal on the deal, assuming the player always choose the suit that already has the most cards to the royal, the probability of completing the royal, and the product.

Royal or Nothing Player

Card to Royal
on Deal
Deal
Probability
Probability
Complete Royal
Product
0 0.61538462 0.00000261 0.00000160
1 0.35444947 0.00003064 0.00001086
2 0.02835596 0.00070472 0.00001998
3 0.00173608 0.01057082 0.00001835
4 0.00007234 0.11627907 0.00000841
5 0.00000154 1.00000000 0.00000154
Total 1.00000000 0.00006075


The lower right cell shows a "royal or nothing" player has a probability of making a royal flush of 0.000006075, or 1 in 16,461.

What would be the probability of a royal flush in video poker if the player could peek ahead at the next cards on the draw?

downtowner

Let's assume that the game picks five random cards for the draw, which will be waiting in a queue for the player to discard. For example, if the player discards three, he will get the next three in the queue. The answer is the same, by the way, if there is a specific card on the deal assigned to each card on the draw. That said, the following table shows the probability of holding each number to the royal on the deal, the probability of completing it on the draw, and the product. The lower right cell shows an overall probability of 0.00006075, which equals 1 in 16,461.

Perfect Peeker in Video Poker

Card to Royal
on Deal
Deal
Probability
Probability
Complete Royal
Product
0 0.61538462 0.00000261 0.00000160
1 0.35444947 0.00003064 0.00001086
2 0.02835596 0.00070472 0.00001998
3 0.00173608 0.01057082 0.00001835
4 0.00007234 0.11627907 0.00000841
5 0.00000154 1.00000000 0.00000154
Total 1.00000000 0.00006075


There is a Caribbean Stud Poker table at the Star City casino in Sydney with a progressive jackpot side bet. It costs $2.50 to play and has the following table table:

  • Royal flush: 100% of jackpot
  • Straight flush: 10% of jackpot
  • Four of a kind: $500
  • Full house: $150
  • Straight: $100


What would be the break-even point on the jackpot?

Anonymous

For the side bet to have zero house advantage, the jackpot would need to reach $578,842.11.

I'm sure you've heard of the guy who claims to know of a bug in Aristocrat slot machines and is offering to tell them about it, for a price. If you knew of such a bug, how would you go about getting the most money out of it?

  • A) Hit a big casino hard with a few max bets, then leave for a few weeks before hitting another big casino. Keep going until they fix the hole or back you off.
  • B) Intersperse medium sized winning bets with small losers all over town and milk the cows for hopefully a longer period of time.
  • C) Form a team and do (A).
  • D) Form a team and do (B).
  • E) Contact the manufacturer and offer it to them for a finders fee + a residual.
  • F.) Other?

Ayecarumba

Yes, I've heard that story. For the benefit of my other readers, here is a Wired link to the story: Meet Alex, the Russian Casino Hacker Who Makes Millions Targeting Slot Machines.

Putting moral issues aside, and assuming getting caught is not a major concern, I would go with choice B. I would have a hard time trusting a team to report winnings honestly and not reveal the secret. Seems to me flying under the radar would be the best choice.

Two travelers decide to eat after a long day of walking. One has five pieces of bread and the other three. They decide to pool the bread together. Before starting to eat a third traveler comes along and asks to join them. They say "yes," figuring eight pieces is enough for three people.

After eating, the third man thanks the other two for the meal. He apologizes for not contributing any food but places eight coins on the table for them to divide as they see fit and then walks off. What is the fair way to divide the eight coins between the two men who did contribute bread? You may assume each man ate the same amount of bread.

Anonymous

Remember that each man also ate bread. Perhaps he shouldn't be rewarded for bread to contributed to the pool that he ate himself.


The traveler who provided five pieces of bread should get seven coins, the one who provided three pieces should get one coin.


Diving eight pieces of bread between three people results in 8/3 pieces eaten by each man. If you deduct from the pool what each man ate himself, then the man with five pieces contributed 15/3 - 8/3 = 7/3 pieces. The man with three pieces contributed 9/3 - 8/3 = 1/3 pieces. So, of the 8/3 pieces eaten by the newcomer, 7/8 of them were contributed by the man with five pieces and 1/8 from the man with three pieces. Thus, the fair thing to do would be to give the man with five pieces 7/8 of the money, or 7 coins. The other man obviously gets the other coin.

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