Please provide detailed solutions to the following problems/exercises (4 problems/exercises x 8 points

each):

1) Consider a deck of 52 cards with 13 hearts, 13 spades, 13 clubs, and 13 diamonds and assume that a

card is drawn at random. Define a random variable X to be 1 if a heart is drawn, 2 if a spade is drawn, and

3 if either a club or a diamond is drawn.

a) List the values of X and their probabilities.

b) What is the probability that X is greater than 1?

c) What is the probability that X is either 1 or 3?

2) A random variable has possible values of 20, 21, 22, 23, and 24 that are equally likely to occur.

a) What is the probability that the random variable is less than 23?

b) What is the mean of the random variable?

c) What is the standard deviation of the random variable?

3) Find the probabilities of the following events for a binomial random variable with n = 10 and p = 0.5.

a) Exactly 2 successes

b) More than 2 successes

c) No more than 2 successes

d) Less than 2 successes

e) At least 2 successes

4) The auto parts department of an automotive dealership sends out an average of 8 special orders daily.

The number of special orders is assumed to follow a Poisson distribution.

a) What is the probability that for any day, the number of special orders sent out will be more than 4?

b) What is the standard deviation of the number of special orders sent out daily?