Question

Consider the following experiment.  Four cards are drawn out of a deck with replacement from a well-shuffled deck of cards.  The card that is drawn out is either a heart or it is not a heart.  After a card is drawn out and recorded it is put back into the deck and the deck is reshuffled.   Construct the binomial probability function for x = 0, 1, 2, 3, 4

P(0) =

P(1) =

P(2) =

P(3) =

P(4) =

Answer 1

Four cards are drawn out of a deck with replacement from a well-shuffled deck of cards.  The card that is drawn out is either a heart or it is not a heart.  After a card is drawn out and recorded it is put back into the deck and the deck is reshuffled.The probability of drawing heart =13/52

Let X be the number of hearts drawn.

Therefore X is independently and identically distributed as binomial distribution with n=4, p=13/52=0.25.

Therefore the binomial probability function for x = 0, 1, 2, 3, 4 is given by

P(0)=0.3164063

P(1)=0.421875

P(2)=0.2109375

P(3)=0.046875

P(4)=0.00390625

### By using R command:

> n=4
> p=(13/52)
> p
[1] 0.25
> dbinom(0,4,0.25)
[1] 0.3164063
> dbinom(1,4,0.25)
[1] 0.421875
> dbinom(2,4,0.25)
[1] 0.2109375
> dbinom(3,4,0.25)
[1] 0.046875
> dbinom(4,4,0.25)
[1] 0.00390625