How many 5 card poker hands can be dealt from a deck of cards? What is the probability of getting a hand with 4 kings? What is probability of getting hand with no kings?

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tiburtius | High School Teacher | (Level 2) Educator

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Since there are 52 cards in a deck, you can deal 10 poker hands (50 cards) and there are 2 cards left in the deck.

If you are asking in how many ways can a poker hand be dealt then your answer is `([52],[5])=(52cdot51cdot50cdot49cdot48)/(1cdot2cdot3cdot4cdot5)=2598960`. 

Probability of getting 4 kings:

Probability of event `A` can be calculated as `P(A)=N_A/N` where `N_A` is number of different occurrence of event `A` and `N` is number of all possible outcomes. So probability of getting 4 kings can be calculated as quotient of number of different hands with 4 kings and total number of different hands which is, as stated above, `([52],[5])`.

Let's now calculate number of different hands with 4 kings:

Since there are only 4 kings in a deck we can't choose anything, we must take them all. For the 5th card in our hand we can chose 1 from 48 remaining cards. Thus there are only 48 hands with 4 kings.

So probability of getting 4 kings in a hand is