| Source | https://www.algoexpert.io/questions/blackjack-probability |
|---|---|
| Difficulty | Medium |
| Category | Click to view 👁️Recursion |
In the game of Blackjack, the dealer must draw cards until the sum of the
values of their cards is greater than or equal to a
target value minus 4. For example, traditional Blackjack uses a
target value of 21, so the dealer must draw cards until their
total is greater than or equal to 17, at which point they stop drawing cards
(they "stand"). If the dealer draws a card that brings their total above the
target (above 21 in traditional Blackjack), they lose (they
"bust").
Naturally, when a dealer is drawing cards, they can either end up standing or busting, and this is entirely up to the luck of their draw.
Write a function that takes in a target value as well as a
dealer's startingHand value and returns the probability that
the dealer will bust (go over the target value before
standing). The target value will always be a positive integer,
and the startingHand value will always be a positive integer
that's smaller than or equal to the target value.
For simplicity, you can assume that the dealer has an infinite deck of cards and that each card has a value between 1 and 10 inclusive. The likelihood of drawing a card of any value is always the same, regardless of previous draws.
Your solution should be rounded to 3 decimal places and to the nearest
value. For example, a probability of 0.314159 would be rounded
to 0.314, while a probability of 0.1337 would be
rounded to 0.134.
Sample Input
target = 21
startingHand = 15Sample Output
0.45 // Drawing a 2-6 would result in the dealer standing.
// Drawing a 7-10 would result in the dealer busting.
// Drawing a 1 would result in a 16, meaning the dealer keeps drawing.
// Drawing with a 16 results in a 0.5 probability of busting (6-10 all result in busts).
// The overall probability of busting is 0.4 + (0.1 * 0.5)
// (the probability of busting on the first draw + the probability of busting on the second).