Probability

11

Hi,

Question:
4 cards are drawn without replacement from a standard pack. What is the probability that they are a mix of Kings and Queens?

I was wondering how to go about this question. I thought this is a permutations problem cause there are different ways you can get a mix of kings and queens.

Any help will be much appreciated!

Reply 1

mqb2766

21

Original post

by sienna2266

Hi,
Question:
4 cards are drawn without replacement from a standard pack. What is the probability that they are a mix of Kings and Queens?
I was wondering how to go about this question. I thought this is a permutations problem cause there are different ways you can get a mix of kings and queens.
Any help will be much appreciated!

Any ideas? Maybe start with just black cards (so 26 cards and 2 kings and 2 queens and you select 2 cards). Then does your logic scale up.

Reply 2

sienna2266

OP

11

Original post

by mqb2766

Any ideas? Maybe start with just black cards (so 26 cards and 2 kings and 2 queens and you select 2 cards). Then does your logic scale up.

Thank you for your message. So to find the probability of getting a mix of kings and queens from 26 black cards without replacement of card, find the probability of drawing a king first and then a queen (2/26 x 2/25).. then find the probability of drawing a queen first and then a king (2/26 x 2/25)... then add them together because it's either king then queen drawn or queen then king drawn...so (2/26 x 2/25) + (2/26 x 2/25). Im pretty sure Ive got it all wrong :/

Reply 3

mqb2766

21

Original post

by sienna2266

Thank you for your message. So to find the probability of getting a mix of kings and queens from 26 black cards without replacement of card, find the probability of drawing a king first and then a queen (2/26 x 2/25).. then find the probability of drawing a queen first and then a king (2/26 x 2/25)... then add them together because it's either king then queen drawn or queen then king drawn...so (2/26 x 2/25) + (2/26 x 2/25). Im pretty sure Ive got it all wrong :/

Sounds about right but not sure how easily it generalised to the larger case. Couple of things,you always have 26*25 on the denominator so we could forget about that until the end and just concentrate on counting the numerator, so how many hands are possible. For the simplified case its 4+4=8 as you say, though counting all the kkkq, kkqk,.... combinations in the larger problem is going to be a bit laborious. Instead could you work out how many cards k|q could be selected on the first pick and then how many on the second without any constraint that there must be a mix. Then correct for the mix constraint, which should obviously give 8 (simplified problem)? This should then generalise, Once you get the result, you should notice that you could also have done a similar nCr argument, by diviiding num and denom by 4!

Alternatively count (nCr) the number of combinations of 1 king, 2 kings, 3 kings (in 4 cards) and combine which seems a bit more complex, though there is an obvious symmetry with the 1 king and 3 kings cases.

(edited 1 year ago)

Reply 4

mqb2766

21

Original post

by sienna2266

Thank you for your message. So to find the probability of getting a mix of kings and queens from 26 black cards without replacement of card, find the probability of drawing a king first and then a queen (2/26 x 2/25).. then find the probability of drawing a queen first and then a king (2/26 x 2/25)... then add them together because it's either king then queen drawn or queen then king drawn...so (2/26 x 2/25) + (2/26 x 2/25). Im pretty sure Ive got it all wrong :/

Did you get it in the end?

Reply 5

Kallisto

Entertainment Forum Helper, Life & Style Forum Helper

22

Original post

by sienna2266

Hi,

Question:
4 cards are drawn without replacement from a standard pack. What is the probability that they are a mix of Kings and Queens?

I was wondering how to go about this question. I thought this is a permutations problem cause there are different ways you can get a mix of kings and queens.

Any help will be much appreciated!

Are the cards drawn one after another? as I got you need at least one king and one queen. Say an ace and the numbers, 7, 8, 9 and 10 plus jack, king and queen exist four times for each, there are 4 * 8 = 32 different cards.

Maybe it is easier to calculate it with complementary probability that contains:

1.) four cards with just one king, but no queen.
2.) four cards with just one queen, but no king.
3.) four cards with either queen nor king.
4.) four cards with two queens, no king.
5.) four cards with three queens, no king.
6.) four cards with four queen.
7.) four cards with two kings, no queen.
8.) four cards with three kings, no queen.
9.) four cards with four kings.

Try this.

For every single draw there are always three possibilities: a king, a queen or another card is drawn.

Good luck.

Probability

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