You have a standard pack of 52 playing cards. You then shuffle them and begin to draw out cards until you have three of a kind. What is the most likely number of cards drawn when this happens?

You then shuffle another pack of 52 playing cards into the pile. What happens to the expected number of cards now? (i.e. does it double / halve / stay the same?)

It is important to use precise language on this website, so people don't spin their wheels. A "three of a kind" in poker, involves 5 cards, not 3. 3 aces, a king and a five are a "three of a kind", but 3 fours and 2 jacks is a full house, not three of a kind. If you draw a king, a king, a king, a queen and a queen in your first five draws, that is not three of a kind. The solution that follows relies on this interpretation.

Part 1:

"You have a standard pack of 52 playing cards. You then shuffle them and begin to draw out cards until you have three of a kind. [It doesn't matter which card this is, so let's just consider the case of 3 aces and 2 nonmatching cards]. What is the most likely number of cards drawn when this happens?"

Answer (believe it or not): 5 cards !!!!!! When you go beyond 5 cards, the probablility of getting a "spoiler" (2 pairs, eg) increases rapidly.

You can only achieve a true "three of a kind" hand by drawing between 5 and 14 cards.

The various probabilities in a 52 card deck are:

There are 52*51*50*49*48=311875200 ways to draw the first 5 cards, of which there are (4*3*2)*(48*44)*(5*4*3*2*1)=2880*(48*44)=6082560 ways to get 3 aces and 2 nonmatching cards. Odds=0.0195031859

There are 52*51*50*49*48*47=14658134400 ways to draw the first 6 cards, of which there are 2880*(48*44*40)=243302400 ways to get 3 aces and 2 nonmatching cards. Odds=0.0165984561

There are 52*51*50*49*48*47*46=674274182400 ways to draw the first 7 cards, of which there are 2880*(48*44*40*36)=8758886400 ways to get 3 aces and 2 nonmatching cards. Odds=0.0129900961

etc......up to:

There are 52*51*50*49*48*47*46*45*44*43*42*41*40*39*38
=5.8601875986e24 ways to draw the first 15 cards, of which there are
2880*(48*44*40*36*32*28*24*20*16*12*8*4*2)=1.4465363954e18 ways to get 3 aces and 2 nonmatchng cards. Odds=0.0000002468

For 52 cards:
Odds of 3 aces and 2 nonmatching cards ("three of a kind"):
(in the first 5 draws) 0.0195031859
(6 draws) = 0.0165984561
(7 draws) = 0.0129900961
(8 draws) = 0.0080827264
(9 draws) = 0.0044087599
(10 draws) = 0.002050586
(11 draws) = 0.0007811756
(12 draws) = 0.0002286368
(13 draws) = 0.0000457274
(14 draws) = 0.00000469
(15 draws) = 0.0000002468

So 5 cards is the most likely.

Part 2.

"You then shuffle another pack of 52 playing cards into the pile. What happens to the expected number of cards now? (i.e. does it double / halve / stay the same?)"

Answer: It stays the same.

As before, you must draw between 5 and 14 cards from the deck of 104.


There are 104*103*102*101*100=11035502400 ways to draw the first 5 cards, of which there are
(8*7*6)*(96*88)*(5*4*3*2*1)
=40320*(96*88)=340623360 ways to get 3 aces and two nonmatching cards. Odds=0.0308661398

There are 104*103*102*101*100*99=1092514737600 ways to draw the first 6 cards, of which there are 40320*(96*88*80)=27249868800 ways to get 3 aces and 2 nonmatching cards. Odds=0.0249423352

There are 104*103*102*101*100*99*98=1.049251154e16 ways to draw the first 7 cards, of which there are
40320*(96*88*80*72*64)=1743991603200 ways to get 3 aces and 2 nonmatching cards. Odds=0.000166213

etc.... up to:

There are 104*103*102*101*100*99*98*97*96*95*94*94*93*92
=7.0173398711e29 ways to draw the first 14 cards, of which there are
40320*(96*88*80*72*64*56*48*40*32*24*16*8)=1.1060024408e23 ways to get 3 aces and 2 nonmatching cards. Odds=0.0000001576


For 104 cards:
The odds of getting 3 aces and 2 nonmatching cards are:
(1st 5 draws) = 0.0308661398
(6 draws) = 0.0249423352
(7 draws) = .000166213
(8 draws) = .000095958
(9 draws) = .000047979
(10 draws) = .0000202017
(11 draws) = .0000068772
(12 draws) = .0000017559
(13 draws) = .0000018125
(14 draws) = .0000001576

So again 5 cards is the most likely number.





Edited on November 20, 2003, 8:05 am

Posted by Dan on 2003-11-20 03:33:26