Consider two identical decks of standard playing cards and use factorial notatio
ID: 3039758 • Letter: C
Question
Consider two identical decks of standard playing cards and use factorial notation where ever possible.
(a) How many distinguishable sequences can result from thoroughly shuffling one deck?
(b) What is the probability that the first card in the single shuffled deck will be an A or a K, irrespective of suit?
(c) What is the probability that the first three cards in the single shuffled deck will be AKQ, irrespective of suit, in that order?
(d) What is the probability that the first three cards in the single shuffled deck will be AKQ, irrespective of suit, in any order?
(e) What is the probability that the first three cards in the single shuffled deck will be AKQ, from a single suit, in that order?
(f) What is the probability that the first three cards in the single shuffled deck will be AKQ, from a single suit, in any order?
(g) What is the probability that the first three cards in the single shuffled deck will be AAA, in any order?
(h) How many distinguishable sequences can result from thoroughly shuffling both decks together?
Explanation / Answer
(a)
A complete deck has 52 cards. Since all the cards are unique, so the number of different possible sequences that can be produced by shuffling the deck = 52!
(b)
There are 4 A and 4 K cards in a deck.
So the probability that the first card will be either an A or a K = (4+4)/52 = 0.154
(c)
There are 4 cards each of A, K and Q.
So the required probability = (4/52)*(4/51)*(4/50) = 0.000482
(d)
If the order AKQ is not important, then the probability calculated above increases by a factor of 6!.
So the required probability = 0.000482*3! = 0.000482*6 = 0.002892
Hope this helps !