A poker hand contains five cards find the variance of the ra

A poker hand contains five cards find the variance of the random variable of

The number of spades in a poker hand

answer is0.86397 please show how to do this

Solution

A Poker hand contanins 5 cards.

Let us consider a random variable X as the number of spades in a poker hand.

And we know that P(X = n) = P ( 5 are spades and 5 - n are not spade)

So the distribution is,

P(X = n) = [ (13 C n) * (39 C 5 - n) ] / (52 C 5) (n is form 0 to 5)

There are total 13 cards of spade.

The non spade cards are 52 - 13 = 39

Total number of cards are 52

We have to make a poker hand of 5 cards.

First we find expectation.

E(X) = n * P(X = n) (n is from 0 to 5)

For x = 0

0 * P(X = 0) = 0

For x = 1

1 * P(X = 1) = 0.41142

For x = 2

2 * P(X = 2) = 0.548559

For x = 3

3 * P(X = 3) = 0.244628

for x = 4

4 * P(X = 4) = 0.042917

for x = 5,

5 * P(X = 5) = 0.002476

Add all these probabilities so that we get E(X).

E(X) = 1.25

Now we find variance ,

variance = [ n - E(X) ]² * P(X =n)

P(X=0) = 0.221534

P(X=1) = 0.41142

P(X=2) =0.27428

P(X=3) = 0.081543

P(X=4) = 0.010729

P(X=5) = 0.000495

For n=0

[ n - E(X) ]² = [ 0 - 1.25]² = 1.5625

[ n - E(X) ]² = [ 1 -1.25]² = 0.0625

[ n - E(X) ]² = [ 2 - 1.25]² = 0.5625

[ n - E(X) ]² = [ 3 - 1.25 ] ² = 3.0625

[ n - E(X) ]² = [ 4 - 1.25 ] ² = 7.5625

[ n - E(X) ]² = [ 5 - 1.25]² = 14.0625

variance = [ n - E(X) ]² * P(X =n)