Question

One of the ways that players can increase their odds of winning at blackjack is by keeping track of what cards have been played, and using that information to decide what to do. To counteract this strategy, casinos usually run games with 6 to 8 decks shuffled together and machines to continuously shuffle the cards, making it all but impossible to count the cards. As a result, when calculating odds for casino games of blackjack we can make the simplifying assumption that the casino is using an "infinite deck," which means that regardless of the cards that have already been dealt, each card (ignoring suits) has a 1/13 chance of being dealt next. This assumption only has a miniscule effect on the odds compared to a real game, but can make calculations much more straightforward.

To give you a more intuitive sense for how close a 6-deck game is to an infinite-deck game, probabilistically speaking, consider the following:

In a game of Blackjack played with 6 decks (312 cards), you look out at the other 20 visible cards on the table and see 4 of the 24 aces.

The casino manager comes over at this point in the game and requests that the dealer shuffle 2 more full decks into the unused cards, bringing the total that you’re playing with up to 416 cards. About how much has the probability of being dealt another ace changed?

A)  ≈1/75 - the probability that a pregnancy results in twins.

B)   ≈1/500 - the probability of being born with 11 fingers or toes.

C)   ≈1/12000 - the probability of getting struck by lightning in your lifetime.

Answer 1

There are 24 - 4 = 20 aces still exists in remaining 312-20 = 292 cards

probability of being dealt another ace = 20/292 = 0.0685

Now if dealer shuffle 2 more full decks into the unused cards, there will be 20 + 8 = 28 aces in 292 + 2 * 52 = 396 cards.

New probability of being dealt another ace = 28/396 = 0.0707

The change in probability of being dealt another ace = 0.0707 - 0.0685 = 0.0022 1/500

B)   ≈1/500 - the probability of being born with 11 fingers or toes.