In a game of blackjack, a 2-card hand consisting of an ace and a face card or a 10 is called a blackjack. (Round your answers to four decimal places.)
(a) If a player is dealt 2 cards from a standard deck of 52 well-shuffled cards, what is the probability that the player will receive a blackjack?
(b) If a player is dealt 2 cards from 2 well shuffled standard decks, what is the probability that the player will receive a blackjack?
(a) P(A,F) = 4/52 * 16/51
P(F,A) = 16/52 * 4/51
so, add them up.
(b) should be clear, but with more cards.
(a) To calculate the probability of receiving a blackjack with a standard deck of 52 cards, we need to determine the number of blackjack combinations and divide it by the total number of possible combinations.
There are 4 aces and 16 face cards (4 each of 10, J, Q, K) in a standard deck. So, the number of blackjack combinations is the product of these two numbers: 4 * 16 = 64.
The total number of possible combinations for receiving 2 cards from a deck of 52 cards is given by the combination formula: C(52, 2) = 52! / (2! * (52-2)!).
Simplifying: C(52, 2) = 52! / (2! * 50!) = (52 * 51) / 2 = 1326.
Thus, the probability of receiving a blackjack is: 64 / 1326 ≈ 0.0483 or 4.83%.
(b) If two decks are well-shuffled together, then the total number of cards becomes 52 * 2 = 104.
The number of blackjack combinations remains the same, which is 64.
The total number of possible combinations for receiving 2 cards from 104 cards is given by the combination formula: C(104, 2) = 104! / (2! * (104-2)!) = (104 * 103) / 2 = 5356.
Therefore, the probability of receiving a blackjack with two well-shuffled decks is: 64 / 5356 ≈ 0.0119 or 1.19%.
To find the probabilities of receiving a blackjack in each scenario, we first need to determine the number of ways a blackjack can occur and the total number of possible 2-card hands.
(a) In a standard deck of 52 cards, there are 4 aces and 16 face cards or 10s (4 of each face card: Jack, Queen, King, and 10). So, there are a total of 4*16 = 64 possible blackjack hands.
The total number of possible 2-card hands can be calculated using the combination formula: C(52, 2) = 52! / (2! * (52 - 2)!) = 1326.
Therefore, the probability of receiving a blackjack is: 64/1326 ≈ 0.0482 (rounding to four decimal places).
(b) When 2 decks are used, the total number of each card doubles. So, there are now 8 aces and 32 face cards or 10s.
The number of possible blackjack hands remains the same since the number of aces and face cards or 10s has doubled. Therefore, there are still a total of 64 possible blackjack hands.
The total number of possible 2-card hands from 104 cards can be calculated using the combination formula: C(104, 2) = 104! / (2! * (104 - 2)!) = 5356.
Therefore, the probability of receiving a blackjack with 2 decks is: 64/5356 ≈ 0.0119 (rounding to four decimal places).