I’ve spent the last few weeks discussing expected value and its importance in making decisions when playing casino games. When playing video poker, your decision should be based on maximizing the expected value.
Believe it or not, this is not the absolute case for all decisions. It is true when the amount you wager is fixed. In video poker, you make an initial wager and you are done wagering. Thus, your total wager is always the same and the goal is to maximize the total units returned.
This is usually the case, but not always the case. In some games, the player has a decision to make about whether or not to wager an additional amount — a bet/check situation, or how much to make that additional wager. For example, while it may not seem like it, the decision to double down in blackjack is actually a bet/check situation. The player can make an additional wager (bet) or he can check and just take a normal hit.
Blackjack is also unique in that when the player makes this additional wager, he must give up something as well — the ability to hit more than one card. In a game like Mississippi Stud Poker, the player can make a 1x or 3x wager (or Fold) but choosing 1x or 3x does not limit any further options.
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Why is this important? If a player could double down in blackjack and still hit as many cards as he wanted to, then the outcome of the hand would not change because he doubles down. He would win or lose each hand regardless of the size of the wager.
But, because he loses the ability to hit more than one card, there will be a few cases where it will change the outcome. Imagine the case where the player has an 11 vs. a dealer 9 up-card. He doubles and draws a 2. He now has 13. If he didn’t double, he could (and should) draw another card. Sure, he might bust. Or he might draw to a 20 and win the hand.
The bottom line is that the win and tie frequency has changed as a result of doubling down. It is quite clear that the win and tie frequency directly impact the expected value of any hand. If the equation for expected value is total units returned divided by total units wagered, then it is quite clear that the total units returned for any hand is the win frequency times two times the units wagered plus the tie frequency times the units wagered.
To put this another way, would you rather play a hand with an expected value of 1.10 that you don’t double down or 1.08 where you do? In the first case you will wager $100 and win $110 back on average for a net win of $10. In the second case you will wager $200 and get $216 back for a net win of $16.
Technically the expected value is a little less for the double down situation, but the net win of dollars is higher. We need to take this into account when it is possible to have different wager amounts for a given scenario.