Evaluating a new side bet
~~~~~~~~~~~~~~~~~~~~~~~~~
A new blackjack side bet introduced in certain casinos goes something like this:
A player is allowed to place a side bet of the amount of his choosing. The player is betting that the first 2 cards in his hand will total 20. If successful, the bet pays 4:1. This is a suckers bet and the correct basic strategy decision for this bet is to never take it. Here's why:
There are 2 ways in which your first 2 cards can total 20: 2 ten valued cards, or an Ace and a 9.
There are 312 cards in our 6 deck shoe, 96 of which are 10 valued cards.
So, the probability of getting a 10 as your first card is:
96/312 = .308
The probability of getting another 10 is
95/311 = .305
So, the probability of both of these happening in sequence is:
.308 * .305 = .0939
As you can see, there is only a 9% chance of this happening.
Considering the Ace-9 hand, there are, again, 312 cards in our 6 deck shoe, 24 of which are Aces and 24 of which are 9s.
The probability of getting an Ace as your first card is:
24/312 = .0769
And getting a 9 after that:
24/311 = .0772
So, the probability of both of these happening in sequence is:
.0769 * .0772 = .00594
Since getting a 9 first and an Ace second also counts as 20, we can double this number to:
.00594 * 2 = .0119
Finally, summing up the 2 probabilities, we see that the probability of totalling 20 on your first 2 cards is:
.0939 + .0119 = .106
Just above 10%.
So, to calculate the expectation for this bet, we'll assume we place a 1 dollar bet:
E = (.106 * 4) + (.894 * -1) = .424 + -.894 = -.47
In other words, for every dollar we place on this bet, we expect to lose 47 cents. That's a 47% house disadvantage! Far worse than, for instance, the insurance side bet. Basic strategy perscribes to never take this bet.
However, it seems reasonable to say that as the deck changes, and that 10 cards are more and more likely to be dealt out, it may become advantageous to take this bet. Let's determine roughly how high the Hi-Opt true count needs to be before this bet becomes advantageous.
The expectation can be defined as:
E = X*4 + ((1-X) * -1)
Where X is the probability of drawing a 20 on your first 2 cards.
Juggling the equation around a bit, we get:
E = X*4 + -1 + X = X*5 - 1
Clearly, in order for this bet to become advantageous, X*5 - 1 needs to be greater than 0. This would mean that X needs to be larger than 0.2.
If we ignore the possibility of the Ace-9 and focus simply on the 2 Ten cards, we see that, roughly, the probability of drawing a 10
out of our deck must be around:
sqrt(.2) = 0.447
If we are prepared to accept this figure we can work out the Hi-Opt true count required to provide such a high probability of drawing 10 valued cards.
We can express our problem as:
Q / 52 = 0.447
Where Q is the number of tens required to make our minimum profitability margin. This assumes that our shoe has been dealt down to exactly 1 deck.
Well, Q is roughly:
Q = 52 * 0.447 ~= 23
This means that the deck would have to have at least 23 ten valued cards in it for our bet to be profitable. In the Hi-Opt count system, this would be
23 - 16 = 7
points off of a balanced deck, so therefore the running count would need to be at least 7. Recall that in the Hi-Opt count system, you need to divide the running count by the number of decks remaining to be dealt to get the true count. We chose our deck such that exactly 1 deck remains to be dealt, so our true count is
7 / 1 = 7
Therefore, in order for this side bet to be profitable, the true count needs to be at least 7.
In conclusion, we see that betting that your first 2 cards total 20 with a payoff of 4:1 is a disadvantageous bet when playing basic strategy and should never be taken. However, we also see that this bet can, in rare circumstances, offer a knowledgable player (who counts cards) an opportunity to make a little extra money.
(C) 2004, Doug Hoyte and Hoyte Blackjack Labs