| The basic blackjack game is dealt from a six-deck shoe. Here is the side bet wager. The side bet costs $1 to play. The payoffs are as follows: $25 if the player's first two cards are unsuited Aces $100 if the player's first two cards are suited Aces $250 if the player's first three cards are unsuited Aces $2500 if the player's first three cards are suited Aces Progressive jackpot if player's first four cards are Aces of the same color As we will see, this is not a good bet at all. Why then spend the time and energy writing about a crummy wager in an out-of-the-way casino? Worse yet, why read about such? There are several reasons. First of all, of course, is to answer the Midnight Skulker's question. Second, there are some interesting calculations involved in the analysis of this bet. Thirdly, the analysis will show how to evaluate games having a progressive jackpot; the calculation for the player and the calculation for the house, though similar, are different. Finally, and this is really interesting, I think that this side bet was poorly planned and could easily be improved. Let me discuss the last item first. Suppose that you are playing this side bet and are lucky enough to be dealt three unsuited Aces as your first three cards. Fine, you have a lock on $250. Now with incredible luck you draw another Ace giving you four Aces of different colors. Your additional prize? Zilch! There is no payout for such a hand; you win the $250 and that's it. How crummy is that? Worse, try this. You are lucky enough to be dealt three suited Aces as your first three cards. Great! You have a lock on $2500. Then with uncanny luck you manage to receive a fourth Ace but of a different color than the three suited Aces. This, as we will see, is a very unlikely scenario. Your additional prize? Again, zilch. How awful! Even if the pay tables were generous (they are not) I would not take this bet simply because of the two events just described. Even though these events are rare, knowing that they could happen to me really annoys me and I'll bet it will annoy others as well. In my opinion, someone should have thought of this when designing this game. Well, let's get to the analysis. To begin with this bet is settled within the first four cards dealt. Moreover, assuming optimum play, the first two Aces will be split (and resplitting Aces is allowed in this game) so there will always be at least four cards dealt. So our analysis is based on dealing 4 cards from a 312-card shoe. (Note: unless some ploppie stands on the Ace pair, the following analysis is valid even if the pair is hit rather than split.) The ordering of the cards is essential here so the total number of possible 4 card hands is 312 x 311 x 310 x 309 = 9,294,695,280 distinct ordered hands. There are 231,264 hands that contain four Aces of different colors. Of these, 5,760 are formed by three suited Aces in the first three cards and an Ace of a different color in the fourth card. This number is obtained as follows. There are four ways to pick the suit of the first three cards and then there are 6 x 5 x 4 ways to pick the three suited first three cards. There remain 12 Aces of an opposite color from the chosen suit so there are 4 x 6 x 5 x 4 x 12 = 5760 such hands. One could also obtain four Aces of a different color by starting with three unsuited Aces of the same color and then choose a fourth Ace of a different color. In such a hand, two of the first three cards must be suited. There are 4 ways to choose the suit of these two cards and then 6 x 5 ways of choosing them in order. There are 6 cards left for the off suit, same color card. These three cards can then be arranged in three ways (ssu, sus, uss). As above, there are 12 Aces of a different color to pick for the fourth card. Hence there are 4 x 5 x 6 x 6 x 3 x 12 or 25,960 such hands. Finally, and I'll leave the calculation to you, there are 199,584 hands that can be formed from 3 unsuited different colored Aces plus any Ace as the fourth card. A word of explanation is due. I write my monthly mathematics article on this site one month in advance of posting. With all of my equations and formulas it is undoubtedly the hardest article for editor John Robison to set up and that is why I give him a month to do so. I should like to say that John does a remarkable job at setting it up. Last month's article (which I wrote in August) promised an article about the mathematics related to comps. I wrote the article that follows on Tuesday, September 18th, one week after the obscene tragedy that struck our country. It is self explanatory. I wrote to both John and Frank and explained that this, my November article for the web site, would not be about mathematics. I had no stomach for writing about the mathematics of gaming under the present circumstances. Gaming mathematics pale in comparison to the real issues on my mind. Both John and Frank made the decision to switch my October and November articles -- thank you guys. [Editor's note: More than just switching the two articles, we decided not to wait until October and to post this article right away.] My heart goes out to all of those who lost their loved ones in this tragic event. I am proud of our firefighters, our policemen, our EMTs, and our politicians of both parties; they have all been wonderful. Nevertheless I am still sad. And Angry! |
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