Many variants of Blackjack allow the player to purchase insurance on their hand. In general, if the dealer’s face up card is an Ace, the player has the opportunity to take insurance. If the player opts to take this action, they must place half of their wager in an area marked “insurance.” If the dealer receives a natural Blackjack, the payout to the player is 2 to 1. If the dealer does not have Blackjack, they lose the insurance bet.
The Insurance Bet is a Losing Wager
In general, the insurance bet is a losing wager. As a result, many choose not to take this and move on with the game. However, there are many mathematically minded Blackjack players that would like to know why it is a losing bet.
The Mathematics of the Insurance Bet
In a single deck game, 3 cards out of the total 52 are face up when the insurance bet is offered. The dealer’s face up card is an Ace so let’s assume the player has not been dealt any 10 value card. This leaves 49 cards in the deck, 16 of which are valued at 10. The probability of winning the insurance bet is 0.327 (16/49). If the bet was $1, then it pays out $2. Therefore the expected payout is 2 * 0.327 = $0.654.
The other 33 cards out of 49 are not valued at 10 so the probability of the insurance bet equates to 0.673. If the insurance bet loses, the player loses his bet of $1 resulting in an expected losing payout of -$0.673. Therefore, the overall net expectation is ($0.654-$0.673) -$0.019. In the long run, the player is expected to lose money because the net expectation is negative.
The player’s expectation actually worsens when dealt 2 10 value cards. Out of 49 cards in the deck, only 14 are worth 10. As a result, the probability of winning is 14/49 which is 0.286. With a $1 insurance bet that pays out $2, the expected payout is $0.572 (2 * 0.286).
35 cards of the 49 in the deck do not have a value of 10 so the probability of losing is 35/49 which is 0.714. Factor in a loss of $1 on the insurance bet and you get -$0.714 with a net expectation of -$0.142. This is considerably worse than the previous example. The numbers prove that the insurance bet is a losing wager and should be avoided at all costs.