Deck of Cards Probability Question

In summary, the probability of getting at least one four-of-a-kind in a thirteen card hand dealt from a standard fifty two card deck is approximately 0.034. The calculation for this probability involves finding the number of combinations for at least one four-of-a-kind and dividing it by the total number of possible combinations for a thirteen card hand. It is important to note that this calculation does not eliminate the possibility of getting multiple four-of-a-kinds in a hand.
  • #1
mr.physics
21
0

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

 
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  • #2
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?
 
  • #3
I am asking for at least one four of a kind.
 
  • #4
I got about 3.43%. What have you tried so far?
 
  • #5
vela said:
I got about 3.43%.

I got this answer as well.
 
  • #6
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?
 
  • #7
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?
 
  • #8
ahh, i see
Thanks!
I'm still not sure how to eliminate the repetitious combinations though...
 
  • #9
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.
 
  • #10
"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.
 
  • #11
Yeah, you're right. I'll have to ponder this a bit.
 
  • #12
This might help:

[tex]\{\texttt{hands with three four-of-a-kinds}\} \subset \{\texttt{hands with at least two four-of-a-kinds}\}[/tex]
[tex]\subset \{\texttt{hands with at least one four-of-a-kind}\}[/tex]
 
Last edited:
  • #13
vela said:
I got about 3.43%. What have you tried so far?

pbandjay said:
I got this answer as well.

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

So you mean .034 approximately.
 

What is a "Deck of Cards Probability Question"?

A "Deck of Cards Probability Question" is a type of mathematical problem that involves calculating the likelihood of certain outcomes when drawing cards from a deck. These questions often require knowledge of basic probability concepts and the structure of a standard deck of cards.

How many cards are in a deck?

A standard deck of playing cards contains 52 cards. These cards are divided into four suits (hearts, diamonds, clubs, and spades) with each suit containing 13 cards (ace through king).

What is the probability of drawing a specific card from a deck?

The probability of drawing a specific card from a deck depends on the number of that card in the deck and the total number of cards in the deck. For example, the probability of drawing the ace of spades from a standard deck is 1/52 since there is only one ace of spades in the deck and 52 total cards.

What is the difference between theoretical and experimental probability?

Theoretical probability is the likelihood of an event occurring based on mathematical calculations and assumptions. Experimental probability is the likelihood of an event occurring based on actual data or results from experiments or trials. In the context of a "Deck of Cards Probability Question", theoretical probability would involve calculating the chance of drawing a specific card from a deck, while experimental probability would involve conducting multiple trials and recording the actual outcomes.

What are some common strategies for solving "Deck of Cards Probability Questions"?

Some common strategies for solving "Deck of Cards Probability Questions" include using tree diagrams, making a list of all possible outcomes, and using combinations and permutations. It can also be helpful to understand the basic rules of probability, such as the multiplication and addition rules, and applying them to the specific question at hand.

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