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+12 votes
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The minimum number of cards to be dealt from an arbitrarily shuffled deck of 52 cards to guarantee that three cards are from same suit is

  1. 3
  2. 8
  3. 9
  4. 12
asked in Probability by Veteran (68.8k points)
retagged by | 1.7k views
use pigeonhole principle Answer is 2*4+1=9
Please re tag this to pigeon hole. I was looking in the GO book under pigeon hole.Its not probability
N=(R-1)*k+1

k = no of pigeon holes

R=expected output here 3

N = minimum number

here we assign an R-1 number of items to k holes and to then add Rth item to any of the holes to get the minimum number

3 Answers

+22 votes
Best answer

There are 4 sets of cards. So, up till 8 cards there is a chance that no more than 2 cards are from a given set. But, once we pick the 9th one, it should make  3 cards from any one of the sets. So, (C) is the answer.

answered by Veteran (14.6k points)
selected by
may be in first 3 attempt we get same(suits) card and question is asking about minimum.

then A (3) is best choice
minimum in worst case.
But it will not guarantee  that three cards are from same suit. Only Picking 9 or more ( 9 or10 or ......or 52) will guarantee that three cards are from same suit.

Minimum of( 9 ,10, 11...52)= 9, So, 9 is the answer(minimum number of cards to be dealt to guarantee that three cards are from same suit)
+6 votes
As suggested above also

apply pigeon hole, 4 holes (suits)

n pigeons(no of cards to be drawn)

floor [(n-1)/p] +1=3

floor[(n-1)/4]  =2

(n-1)/4  >= 2

n>=9

minimum 9 cards must be picked
answered by Boss (8.5k points)
+3 votes

Can this question be solved by calculating expectation:
Expectation of getting three cards from same deck->

E(x) = 13*(13/52) + 12*(12/51) + 11*(11/50)
       =  8.49 => 9 cards

Hence is the answer.
Is it correct?

answered by Boss (6.1k points)
@shikhar May you please explain your approach more? I find it's correct use of "Expectation".


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