7 standard decks of 52 cards are shuffled together, resulting in a deck with 364

Question

7 standard decks of 52 cards are shuffled together, resulting in a deck with 364 cards. You are dealt two cards out of this deck. Determine the probability of a "21." In other words, determine the probability that one of your cards is an Ace (A) and the other card has face value 10 (one of the cards 10, J, Q, or K) _______? Determine the probability that the two cards are identical. (E.g both cards are the "Queen of Hearts", etc.) _______? Determine the probability that both cards are "Diamonds." _________?

Explanation / Answer

Question 1:

There are 28 aces and 28 tens, so the probability of getting one of each is:

(28C1)(28C1) / (364C2) = 784/66066 = 56/4719 (about 1.187%)


Question 2

For any given card (like the 2 of hearts, for example), there are 7C2 = 21 ways to choose two of them. Multiply this by 52 to get the total number of combinations:

21*52 / 66066 = 1092/66066 = 2/121 (about 1.653%)


Question 3

Probability that both cards are diamonds:

There are 13*7 = 91 diamonds.

(91C2) / (364C2) = 4095 / 66066 = 15/242 (about 6.2%)

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