Question

Find the probability of the following card hands from a 52-card deck. In poker,

aces are either high or low. A bridge hand is made up of 13 cards.
In poker, four of a kind (4 cards of the same value)
O 2.40 x 104
O 2.00 × 10-5
O 1.85 x 10-5
O4.34 x 10-4

Answer 1

Answer: 2.40 * 10^(-4)

This value is approximate.

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Step-by-step explanation:

A four of a kind is where we have 4 cards of the same value, as mentioned in the instructions. For example, we could have 4 queens and some other card, such as a 10 of clubs.

There are 13 cards to pick from for any given suit. Once that card is chosen, the other 3 slots are fixed to whatever that first card is. The fifth slot will have 52-4 = 48 choices.

In short we have 13 choices for the first four cards and 48 choices for the fifth card.

Overall we have 13*48 = 624 different four of kind poker hands.

Let's now calculate how many possible poker hands there can be.

A poker hand consists of 5 cards. I don't know why your teacher mentioned a bridge hand of 13 cards since that's for a different game.

We'll use n = 52 and r = 5 in the nCr combination formula below.

`n C r = (n!)/(r!(n-r)!)\\\\52 C 5 = (52!)/(5!*(52-5)!)\\\\52 C 5 = (52!)/(5!*47!)\\\\52 C 5 = (52*51*50*49*48*47!)/(5!*47!)\\\\ 52 C 5 = (52*51*50*49*48)/(5!)\\\\ 52 C 5 = (52*51*50*49*48)/(5*4*3*2*1)\\\\ 52 C 5 = (311875200)/(120)\\\\ 52 C 5 = 2598960\\\\`

There are 2,598,960 possible five-card poker hands.

624 of those hands are four of a kind, so the probability we want is

624/(2,598,960) = 0.000240 = 2.40 * 10^(-4)

The approximate decimal value 0.000240 converts to 0.0240%