Insurance is one of the least understood rules in blackjack. It's essentially a side-bet that protects players in the event of a dealer blackjack. If the dealer's upcard is an Ace, players will be offered a chance to buy insurance up to half of their original bet.
Once insurance is bought, the dealer checks to see if he has a ten-valued card in the hole. There are two possible results:
If the dealer has blackjack, and the maximum allowable amount of insurance is purchased, the player breaks even. The original bet is protected, hence the name "insurance."
So are insurance bets worth the money? No, it is not.
To calculate the odds, we look at insurance strictly as a side-bet. In order for the insurance bet to break even, the hole card must have a value of ten at least 1 in 3 times. Anything less than 1 in 3 will result in a net loss over time.
In a single-deck game, there are 16 cards with a value of ten. Removing the dealer's Ace leaves 51 cards, which we will assume are all unknown for simplicity. Thus, there is a 16/51 chance that the hole card is a ten. This equals 1/3.1875 (31.4%), which is less than the required 1/3 (33.3%). In this case, buying insurance is not a cost-effective option.
Having said that, there are situations in which buying insurance may be worthwhile. The odds of the dealer having a ten-valued card in the hole are equal to the percentage of ten-valued cards remaining. The exact odds for any single hand can be calculated using the following formula:
[(N x 16) - A] / [(N x 52) - (A + B)]; where N = Number of Decks, A = Tens Dealt, and B = Non-Tens Dealt.
If the result is less than 1/3, insurance should not be purchased. For example, if you are the only player at the table and you are dealt two face cards in a single-deck game of blackjack, the odds would be [(1 x 16) - 2] / [(1 x 52) - (2 + 1)] = 1/3.5. Obviously, this is less than 1/3 (28.6% vs. 33.3%, respectively) and an insurance bet would not be mathematically justified. On the other hand, if another player joined you at the table, and neither hand had a ten-valued card, the odds would be [(1 x 16) - 0] / [(1 x 52) - (0 + 5)] = 1/2.9375 (34.0%). In this situation, it would be worthwhile to buy insurance.
Card counters keep track of the number of ten-valued cards that have been played, and can determine when more than one-third of the remaining cards are ten-valued. This instance is rare and is the only situation in which purchasing insurance is profitable. Unless you are able to count cards and ensure a favorable bet, blackjack insurance is best avoided.
The situation changes slightly when you are dealt a blackjack. In this case, the dealer may offer "even money" instead of the normal insurance bet. Even money is basically a simplified form of insurance. Suppose you bet $20 and have a blackjack. You would usually win $30 for this, unless the dealer has a blackjack, too, in which case the hand would result in a push.
If the dealer's upcard is an Ace, and you decide to insure your blackjack for the full amount, $10 in this case, two things can occur:
Either way, the player is guaranteed a profit of $20, or even money for the original bet. Casinos thus eliminate the insurance bet altogether and allow the player to pronounce even money for a blackjack when the dealer shows an Ace.
A guaranteed profit in the face of a dealer blackjack may sound like a good deal, but it isn't. You are sacrificing one-third of your potential profit to protect against something that occurs less than one-third of the time. You will win more money in the long run if you hold out for the $30, even though you will lose it all occasionally.
In general, it is not wise to take an insurance bet, including the "even money" variant. The only time it is profitable is when over one-third of all remaining cards are ten-valued. Unless you know this to be the case, ignore blackjack insurance.