30 points) Consider chance experiments of fairly drawing cards from a standard 5

Question

30 points) Consider chance experiments of fairly drawing cards from a standard 52-card deck. Recall a rd card deck has 4 suits (spades, clubs, diamonds and hearts) and the possible denominations for each suit are: 2, 3, 4, 5, 6, 7, 8,9, 10, J, Q, K, A. We can view the 52 cards in the following table: 2 Spades2s 3s 4s 5s 6s 7s 8s 9s 10s Js QsKs As Consider the game of drawing 5 (five) cards (without replacement) from the above standard 52-card set. Determine the number of possible unique "hands" with the given properties and give examples. a) Type A hands: this type of hand has exactly 3 of the four cards with the same denomination and the other 2 cards also have the same (obviously different) denomination b) Give 5 example of Type A hands as defined in part a) (use set notation) c) Repeat a) but this time count the number of ordered hands that are possible. That is, order matters and re-ordered versions of the same hand are counted multiple times. Thus, each outcome is an ordered set of 5 cards. JUSTIFY

Explanation / Answer

a)

To find the favorable number of ways we will first select one denomination out of given 13 and then select 3 cards out of the 4 cards belonging to then denomination. After that we will select one denomination out of the remaining 12 denominations and then select 2 cards out of the 4 cards of that denomination. Therefore the favorable number of ways would be given by,

N = C(13,1)C(4,3)C(12,1)C(4,2) = 3744

b)

S = {2S2D2H3S3D , KCKDKH3S3D , ASADAHQSQC , 9S9D9H6S6H , QDQCQH7C7H}

c)

In case the ordering matters we will just multiply the number of favorabe hands with 5! to account for the ordering of these cards. Hence the required numbers would be given by,

N = 3744*5! = 449280

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