How many 5 card hands are possible that have all 4 aces?

  • Apr 10, 2012
  • #1

Unite1133

10

If you have a standard deck of 52 cards, what is the probability that out of a hand of 5 cards you get 4 aces?



First I found the total # of ways for choosing 5 cards from 52 = (52 C 5) = 2,598,960
Then the # of hands which has 4 aces is 48 (because the 5th card can be any of 48 other cards).
So there is 1 chance in (2,598,960/48) = 54,145 of being dealt 4 aces in a 5 card hand.
The probability is 1/54,145 ≈ .0018469%

Did I do this right?

 

  • Apr 10, 2012
  • #2

Dick

Science Advisor

Homework Helper

26,263621

Looks ok to me.

 

G

Guest

Guest

  • May 31, 2006
  • #1

Hello, i am totaly lost on this problem and have already been reduced to punching randome numbers in the calculator. Could someone please tell me how to start this problem?

What is the probability of holding all 4 aces in a 5 card hand dealt from a standard 52 card deck?

Thanks

 

How many 5 card hands are possible that have all 4 aces?

pka

Elite Member

JoinedJan 29, 2005Messages11,799

  • May 31, 2006
  • #4

If a five-card deal has all four aces then how many ways can there be a fifth card?

 

G

Guest

Guest

  • May 31, 2006
  • #5

52-4? So that would give you the 48 right?

Sorry if i sound dumb but its hard to think straight when panicking, the test on this is friday. :cry:

 

S

soroban

Elite Member

JoinedJan 28, 2005Messages5,586

  • Jun 1, 2006
  • #7

Re: cards: probability of holding all 4 aces in a 5-card han

Hello, adon!

Okay, some baby-talk may be in order . . .

What is the probability of holding all 4 aces in a 5-card hand dealt from a standard 52-card deck?

Click to expand...

First, there are \(\displaystyle \,\begin{pmatrix}52\\5\end{pmatrix}\,= \,2,598,960\) possible hands.

Now, how many 5-card hands will contain the four Aces?

There is only 1 way to have the four Aces.
The fifth card can be any of the remaining 48 cards.
\(\displaystyle \;\;\)Hence, there are: \(\displaystyle \,1\,\times\,48\:=\:48\) hands that contain the four Aces.

Therefore: \(\displaystyle \,P(\text{4 Aces})\;=\;\L\frac{48}{2,598,960}\;=\;\frac{1}{54,145}\)

 

What is the probability that a five-card poker hand has four ACES? When I was solving the above stated problem, I got confused while trying different methods :

Assume a normal $52$ deck of cards.

Method 1:

Selecting the $4$ aces from total $4$ aces can be done in $\mathsf C(4,4)$ ways and selecting any non ace element from rest $48$ cards can be done by $\mathsf C(48,1)$ ways. Any $5$ cards can be drawn from $52$ deck of card in $\mathsf C(52,5)$ ways. So the probability is $$\frac{\mathsf C(4,4)\times \mathsf C(48,1)}{\mathsf C(52,5)}$$


Method 2:

We have $4$ aces in total. so probability of selecting an ace from $52$ cards is $4/524$ , then we are left with $51$ cards and selecting again another ace gives probability $3/51$. Similarly for next two aces probability will be $2/50$ and $1/49$. Now we are left with total $48$ cards and we can obviously choose any of these $48$ card which gives probability of $48/48$. Multiplying the probabilities gives us $$\frac{(4\cdot 3\cdot 2\cdot 1\cdot 48)}{(52\cdot 51\cdot 50\cdot 49\cdot 48)}.$$

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How many 5 card hands have 4 aces?

If you have a standard deck of 52 cards, what is the probability that out of a hand of 5 cards you get 4 aces? Then the # of hands which has 4 aces is 48 (because the 5th card can be any of 48 other cards). So there is 1 chance in (2,598,960/48) = 54,145 of being dealt 4 aces in a 5 card hand.

What is the probability of getting 4 of a kind in a 5 card hand?

FOUR OF A KIND The probability is 0.000240.

How many 5 card hands will have 4 aces and 1 king?

36. the 4 aces plus 1 card from 2-k(12 ranks) from diamonds hearts and spades.

How many possible 5 card hands are there?

First, count the number of five-card hands that can be dealt from a standard deck of 52 cards. We did this previously, and found that there are 2,598,960 distinct poker hands.