3. Each card in a standard 52 card deck has both a suit\" and a \"rank\". There

Question

3. Each card in a standard 52 card deck has both a suit" and a "rank". There are 4 suits, which are spades, hearts, diamonds, clubs. There are 13 ranks, 2,3,..., 10, J, Q, K, A. (a) How many different 5 card hands, drawn from a standard 52 card deck, have a two-pair given by having 2 Jacks and 2 Aces? (Note that weqwant to exclude the possibility of having a full-house, which is a hand with 2 of one rank and 3 of another, here). (b) Given a standard 52 card deck, what is the probability of drawing a two-pair (and not a full-house) given by having 2 Jacks and 2 Aces if you draw a 5 card hand?

Explanation / Answer

a). We have a five hand containing 2 aces, 2 jacks and 1 other card. now we need to find the total number of different ways in which this can be done.

Clearly n things can be arranged in in n place in n! ways

Tus 5 cards can be rearranged in 5 hands in 5! ways =120 ways.

b). As per the question following occurs.

1. 2 cards form jack (without order)

2. 2 cards of aces (witohut order)

3. one card from the set which is neither jack nor ace (from 48 left out cards)

4. replacement of cards is not allowed.

Let us draw first two cards as jack then two cards aces then other card

probability of first card being jack =4/52 (favourable cards/total cards)

when 2nd card is drawn then favorable case is 3 (one is already taken out) and total cases is 51.

Thus probability 3/51

simlilarly we can find probability for all other cards.

Hands 1 2 3 4 5 Cards Jack Jack Ace Ace Other Probabilty 4/52 3/51 4/50 3/49 44/48 0.08 0.06 0.08 0.06 0.92 Probabilty 0.0000203

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