Question

A special deck of 45 cards has 15 cards with a picture of a ghost, 15 cards with a picture of a goblin, and 15 cards with a picture of a witch.

a. 8 cards are picked in a row without replacement from this special deck. Let X be the number of cards which are picked and have a picture of a ghost. Find E(X) and V ar(X).

b. Repeat part a, but suppose 20 cards are picked instead of 8.

c. Repeat part a, but suppose 45 cards are picked instead of 8

Answer 1

Hypergeometric Distribution.. Suppose a population consists of N items, k of which are successes. And a random sample drawn from that population consists of n items,

If X is random variable of number of successes drawn from that sample;

Then Probability Mass function of X : P(X=x)

Number of cards in the special deck : N = 45; Population consists of 45 cards

Number of cards with a picture of Ghost = 15 ; k = 15 ; Number of successes

Number of cards picked in a row without replacement from this deck = 8 ; n : Sample of 8 cards drawn

X : Number of cards which are picked and have a picture of Ghost.: X: Number of successes (Ghost ards)

X follows a Hypergeometric Distribution :P(x)

a .

Number of cards picked in a row without replacement from this deck = 8 ; n : Sample of 8 cards drawn

Mean of a Hyper Geometric distirbution = E(X) =

Number of cards in the special deck : N = 45; Population consists of 45 cards

Number of cards with a picture of Ghost = 15 ; k = 15 ; Number of successes

Number of cards picked in a row without replacement from this deck = 8 ; n : Sample of 8 cards drawn

i.e n= 8 ; N=45 ; k=15

E(X) = 2.67

Var(X) = 1.4949

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b.

b .

Number of cards picked in a row without replacement from this deck = 20 ; n : Sample of 20 cards drawn

Mean of a Hyper Geometric distirbution = E(X) =

Number of cards in the special deck : N = 45; Population consists of 45 cards

Number of cards with a picture of Ghost = 15 ; k = 15 ; Number of successes

Number of cards picked in a row without replacement from this deck = 20 ; n : Sample of 20 cards drawn

i.e n= 20; N=45 ; k=15

E(X) = 6.67

Var(X) = 2.5253

c .

Number of cards picked in a row without replacement from this deck = 45 ; n : Sample of 45 cards drawn

Mean of a Hyper Geometric distirbution = E(X) =

Number of cards in the special deck : N = 45; Population consists of 45 cards

Number of cards with a picture of Ghost = 15 ; k = 15 ; Number of successes

Number of cards picked in a row without replacement from this deck = 45; n : Sample of 45 cards drawn

i.e n= 45; N=45 ; k=15

E(X) = 15

Var(X) = 0