Question

1.Suppose that you randomly draw one card from a standard deck of 52 cards. After writing down which card was drawn, you replace the card, and draw another card. You repeat this process until you have drawn 18 cards in all. What is the probability of drawing at least 5 diamonds?

2.For the experiment above, let ? X denote the number of diamonds that are drawn. For this random variable, find its expected value and standard deviation. ?(?)= σ =

Answer 1

1.

Here, we use Binomial distribution

PMF for binomial random variable is :

where,

n = Number of trials

x = Number of trials

p = Probability of success

In our case,

n = 18

p = Probability of diamond card on a single draw = Number of diamond cards / Total cards = 13 / 52 = 1 / 4

Now,

Probability of drawing at least 5 diamonds = P(X 5)

We can write :

This implies :

Now, we use Excel function " BINOM.DIST()" function

So,

Hence,

Probability of drawing at least 5 diamonds = 0.4813

2.

Expected value of X :

Standard deviation of X :