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

Answer:

E[X]=1.5294

and

Var[X].=1.3547

Solution:

We have given

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

We will define

p(success)=p(getting two cards the the same face value )

for any number we have 4 cards with same face value out of 52 select 2 card from these 4 card have probabilit

=4/52*3/51

=12/(52*51)

and we have such 13 different No hence we will multiply above value by 13

=13*12/(52*51)

=0.05882

p(success)=0.05882

Let X be the number of people whose two cards have the same face value

so its nothing but # of success out of 26

hence it follows binomial distribution ( with p=0.05882 and n=26)

X~Binomial(p=0.05882 , n=26)

E[X]=n*p=26*0.05882=1.5294

Var[X]=n*p*(1-p)^2

=1.3547

Answer

E[X]=1.5294

and

Var[X].=1.3547