Expectations Galore: Insurance

There are certain kinds of wagers for which the player has a choice of either making a wager or declining to make a wager, and game conditions change so that either decision may be correct depending on circumstances. In such a case, a player's decision should be made on the basis of whether or not the wager has positive expectation. It is as simple as that, and yet the general population's unwillingness to understand expectation often leads people to adopt silly positions.

The general population's unwillingness to study basic probability is not the only problem. Sometimes there are so-called experts giving false information. The latest example of this I have seen is someone giving false information on the insurance bet in blackjack in a publication produced by the Great Canadian Casino chain in British Columbia. Let's take a look at how the insurance bet should be analyzed.

I have heard many people giving advice on the insurance bet and the advice is usually incorrect. Sometimes it is blatantly incorrect and frequently the discussion is emotional. This reminds me of a recent scene at a casino in Vancouver.

I walk in the door and observe Bib Ladder and Willy the Hummer (I shall not go into the derivation of Willy's nickname other than to say it has nothing to do with the possession of a 95-mile an hour fastball) involved in an animated discussion. Ignoring an inner voice to stay away, I approached the two of them since it had been a long time since I last saw Bib. As I approached, I could see Bib was exasperated.

``Hi you two old guys. What's the great debate all about?''

``Willy here is giving me a lot of rubbish about when to make an insurance bet and...'' at which point Willy interrupted and protested, ``I just saw Bib take insurance when he was holding a 12 ... My God! A 12 ... I couldn't believe it!''

``Hold on, you two. Let's discuss this calmly.''

``Alright, professor, let's hear what you have to say,'' said Willy.

I proceeded to ask Willy when he thought a player should take insurance and heard the spiel about insuring blackjacks, 20s and a few 19s. I figured it was time to introduce Willy to one of the procedures mathematicians sometimes use to try to get a handle on a problem.

I started, ``Let me create two scenarios. First, suppose you are dealt a 20, the dealer has an ace showing and you know your two 10s are the last two 10s left in the shoe. Don't answer me now but would you take insurance?

Second, suppose you are dealt a 13, the dealer has an ace and you know every card left in the shoe is a 10. Would you take insurance?''

``Now let me change the situation a bit. Suppose I cover your cards so you cannot see them and you know that either (a) every card left is a 10 or (b) there are no 10s left and the dealer has an ace. What would you do now with regard to insurance?''

``Of course I would take insurance if all the cards are 10s, and not take it if there were no 10s left, but these situations never arise,'' protested Willy.

I saw Bib smiling as I said, ``You are right, Willy, such situations are unrealistic, especially under conditions in which blackjack is now dealt, but that's not the reason I described two scenarios. You see, when mathematicians are struggling to understand what is involved in a problem on which they are working, sometimes they consider extreme examples as a way of clarifying what the important issues are in the problem. Tell me, Willy, what do these scenarios suggest to be the important considerations?''

Willy stared at me for several seconds apparently unable, or unwilling, to articulate any thoughts. Finally, Bib quietly said, ``The examples clearly illustrate two facts: The number of 10s left in the deck is what should determine whether or not you should bet, and the cards you hold are irrelevant.''

``Bib is correct that these extreme scenarios suggest what he has stated. Mind you, they only suggest it. It is now up to us to see that it remains the case for all situations. Let's begin.''

``When you take the insurance bet, which cannot exceed one-half of your original bet on the hand, the house is offering us 2-to-1 if the dealer has a blackjack. The outcome of the insurance bet has no effect on the outcome of your original bet. That will be decided after the dealer turns over her hand and follows the fixed rules for completion of her hand. Thus, the insurance bet is a side bet on whether or not the dealer has a blackjack. It has nothing to do with your hand.''

``Since it is a side bet, the player should make the bet only if the player has positive expectation. Since the house is offering 2-to-1, the player has positive expectation exactly when the probability of a 10 being dealt to the dealer is greater than one-third. This probability is greater than one-third precisely when more than one-third of the cards left in the shoe are 10s. So if the player is counting 10s and knows precisely how many are left, the decision of whether or not to take insurance is a no-brainer.''

Willy is a stubborn cuss and mumbled something about the possibility I was correct but Bib and I later concurred he probably was going to find it hard to change his ways.

Shakespeare was not always correct no matter how beautifully his thoughts were phrased. When he wrote `that which we call a rose by any other name would smell as sweet', he had no contact with modern spin doctors. I believe the person who gave the name ``insurance'' to the side bet discussed above made a great move for the casinos. The use of the term ``insurance'' has deceived many people into believing there is a connection between the bet and protecting the players hand. If the bet had been called something like ``blackjack side bet'', I suspect more people would think about it correctly. The point is, the insurance bet is handled correctly simply based on the concept of expectation.

As a final aside, those people who claim a player should insure a 20 are not thinking about what they are saying. In the old days, when most games were dealt from a single deck, a player possessing a hand with two 10s had removed one-eighth of the 10s from the deck thereby non-trivially reducing the chances of the deck having enough 10s in it to make the insurance bet have positive expectation. Even with double deck games, a player with two 10s who sees one or more 10s in other players hands is probably not making the correct decision in automatically insuring his hand.

Keeping exact count of the 10s is not the top priority for most counting systems so I doubt many players actually know how many 10s are left. There are criteria one can use but these are approximations. Nevertheless, these approximations together with knowledge of how many aces have gone are useful. (If the deck is rich in big cards and many aces have gone, then there are proportionally more 10s and so on.)