Ask the Slot Expert: Drawing a royal flush

Question: Last night I played a Top Dollar machine with an extra bonus feature. It was $5 for the max credits. I got a Top Dollar and accepted the second offer of $115. Then the second bonus threw up an offer of $15. The casino host told me that if I declined that offer it would reset the first bonus, but it looked to me like the first bonus was locked in. Do you have any insight into these rules?

Answer: Last year IGT released another machine in the Top Dollar theme, Top Dollar Premium. When you bet three coins and trigger the Top Dollar Bonus, you get four chances to accept an offer. If you bet five coins, however, the Top Dollar Extra Prize Bonus gives you five chances to accept two offers.

The page on the IGT website says that this bet structure encourages players to increase their bet from three credits (the minimum required for the bonus) to five credits, due to the "incentives of an extra award and higher winning potential."

I wasn't able to find one of these machines at my usual casinos, so I wasn't able to check the help screens or try it out myself. I'll have to rely on the description on the IGT site.

I think you're right. According to the marketing verbiage, you get five chances to accept two offers. Once you accept an offer, it's locked in and yours. You then have however many offers you have remaining to accept your second offer.

I don't know why your host thought declining the first offer for the second bonus would reset the first bonus. Let's consider the implications. If it did, then you are forced to accept two offers in a row. Another problem is that the game gives you only five offers. If you accepted the third offer and declined the fourth, would the game then have to keep giving you offers until you accepted two in a row?

Well, it doesn't matter. You can accept any two of the five offers you're given in this game that "brings a dazzling new play experience to a classic hit theme." I hope the experience isn't so dazzling that you need to wear your eclipse glasses.

Question: I was dealt a royal on the redraw after discarding all five cards. How does that affect the odds? My instinct is not by much, 600,000 to 1? But boy was I surprised when the machine started racking up credits.

Answer: You might be surprised to see the effect that removing five cards has on the number of hands that can be dealt.

With a full, 52-card deck, there are a total of 2,598,960 five-card hands that can be dealt. Four of them are royal flushes. The chances of being dealt a royal flush are 4 in 2,598,960, or 1 in 649,740.

Let's look at how we calculate the number of hands that can be dealt. We have 52 choices for the first card, 51 for the second, 50 for the third, 49 for the fourth, and 48 for the fifth. Multiplying the values together gives us the number of hands if order mattered (see: permutation). Order doesn't matter in almost all video poker pay tables, so we need to find the number of unique combinations of cards (see: combination). We're drawing five cards, so we need to divide by 5 times 4 times 3 times 2 times 1, or 120.

Now let's look at the calculations after you've discarded five cards. We have 47 choices for the first card, 46 for the second, 45 for the third, 44 for the fourth, and 43 for the fifth. We're still going to divide by 120 because we don't care about the order.

You can see that 47x46x45x44x43 is going to be much less than 52x51x50x49x48 and the number of hands in the 47-card deck also much less. After removing five cards from the deck, there are 1,533,939 five-card hands left in the deck, a bit more than half the number of hands in the full deck. Removing five cards really eliminated a lot of hands.

Your chances of drawing a royal depends on how many royals are still left in the deck. In Jacks or Better, you hold a single high card when that's the best play, but you would never hold a lone 10. In wild-card games, you might not hold a single high card and maybe not even a 2-card royal with an ace.

If all four royals are possible, your chances are 4 in 1,533,939 (1 in 383,484.75). Just three left, 3 in 1,533,939 (1 in 511,313). Two, 2 in 1,533,939 (1 in 766,969.5), and just one, 1 in 1,533,939.

If none of your discards are honor cards or 10s, then you're almost twice as likely to draw a royal than get dealt a royal. Your chances go down when you eliminate possible royals by discarding the cards you need for them.