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 19 cards in all. What is the probability of drawing at least 8 clubs? For the experiment above, let X denote the number of clubs that are drawn. For this random variable, find its expected value and standard deviation.

The variable X follows a binomial distribution, because we have n identical and independent events (19 cards) with a probability p of success and 1-p of fail (there is a probability of 1/4 to be club and 3/4 to be diamond, heart or spade). Then, the probability that x of the n cards are club is:

[tex]P(x)=\frac{n!}{x!(n-x)!}*p^{x}*(1-p)^{n-x}\\P(x)=\frac{19!}{x!(19-x)!}*0.25^{x}*(0.75)^{19-x}[/tex]

So, the probability P of drawing at least 8 clubs is:

P = P(8) + P(9) + P(10) + ... + P(18) + P(19)

Replacing, the values of x, from 8 to 19, on the equation above, we get:

P = 0.0775

Additionally, the expected value E(x) and standard deviationS(x) for this distribution is given by:

E(x)=np = 19(0.25) = 4.75

[tex]S(x)=\sqrt{np(1-p)} =\sqrt{19(0.25)(0.75)} =1.8875[/tex]