Question

suppose a card is selected from a standard deck, its suit is noted, the card is returned to the deck and the deck is shuffled. this process is done 12 times. let X count the number of spades obtained.
a. what kind of discrete random variable is X?
b. what is the p.m.f of X?
c. what is the expected value of X?
d. what is the variance of X?

Answer 1

A card is randomly selected from a standard deck; its suit is noted, and the card is returned to the deck; the deck is reshuffled.

The process is done 12 times.

Now, first of all, there are 13 spades.

So, the chance that a randomly selected card is spade, is 13/52, or 0.25.

This probability remains the same all through the 12 trials, because the cards are replaced, and the deck is reshuffled.

So, if X be the random variable denoting the number of spades, drawn in 12 trials, then X follows binomial distribution with parameters n=12 and p=0.25.

Question a

X is a binomial random variable with parameters n = 12 and p = 0.25.

Question b

The PMF of X is given by

where

Question c

The expected value of X is

Question d

The variance of X is