Question

A standard deck of 52 cards is split evenly among 26 people. Let X be the number of people whose two cards have the same face value (e.g., {9♠, 9♥}). Calculate E[X] and Var[X].

Answer 1

We have 4 sets in a 52 cards. Each set have 13 similar cards.( A,2,3,4,5,6,7,8,9,10,K,J,Q)

P= 2 cards have the same face value

n= selecting 2 cards from 52 cards as

= 52c2= 1326

m= selecting 2 cards have the same face value

= in 4 sets selecting 2cards as 4c2= 6ways

In similar way it can form 13 similar groups

m= 13×6 =78

P= m/n= 78/ 1326= 0.0588

Given data are sample n =26 people

X= number of people whose 2 cards have the same face value

P= 0.0588

q= 1-p= 1-0.0588 =0.9412

Mean= E [X]= np = 26× 0.0588= 1.5288

Variance V [X]= npq

= 26× 0.0588 × 0.9412= 1.4389