Deck of Cards Probability Question

[mr.physics]

Homework Statement

What is the probability of gettting four-of-a-kind in a thirteen card hand dealt from a standard fifty two card deck?

Homework Equations

The Attempt at a Solution

 

[pbandjay]

Are you asking for the probably at least one four of a kind? Or is having multiple four of a kinds in the hand illegal?

 

[mr.physics]

I am asking for at least one four of a kind.

 

[vela]

I got about 3.43%. What have you tried so far?

 

[pbandjay]

I got about 3.43%.

I got this answer as well.

 

[mr.physics]

I got .0753 = ((13)(52C9))/(52C13) although i suspect that there are some repetitious combinations ie. 4,4,4,4,5,6,7,7,7,7,5,6,2 and 7,7,7,7,5,6,4,4,4,4,5,6,2.

Could you explain how you arrived at your answer?

 

[vela]

Your mistake is in the numerator. After you draw the four of a kind, how many cards are left from which to draw the remaining 9 cards?

 

[mr.physics]

ahh, i see
Thanks!
I'm still not sure how to eliminate the repetitious combinations though...

 

[vela]

You haven't eliminated repetitions like getting two or three sets of four of a kind, but that's okay because you said you wanted the probability of at least one four of a kind, not exactly one four of a kind.

 

[mr.physics]

"You haven't eliminated repetitions like getting two or three sets of four of a kind"

In calculating the combinations for any particular four-of-a-kind, for example for the combinations of the four-of-a-kinds for cards 4 and 7, there will be one combination that is counted twice: 4,4,4,4,5,6,7,7,7,7,5,6,2 and 7,7,7,7,5,6,4,4,4,4,5,6,2.

 

[vela]

Yeah, you're right. I'll have to ponder this a bit.

 

[vela]

This might help:

$$\{\texttt{hands with three four-of-a-kinds}\} \subset \{\texttt{hands with at least two four-of-a-kinds}\}$$
$$\subset \{\texttt{hands with at least one four-of-a-kind}\}$$

 

[LCKurtz]

I got about 3.43%. What have you tried so far?

I got this answer as well.

I agree, but remember probabilities are numbers between 0 and 1, not percentages.

So you mean .034 approximately.