Question

Suppose that nine Massachusetts athletes are scheduled to appear at a charity benefit. The nine are randomly chosen from eight volunteers from the Boston Celtics and four volunteers from the New England Patriots. We are interested in the number of Patriots picked.

a. In words, define the random variable X.

b. List the values that X may take on.

c. Give the distribution of X. X ~ _____(_____,_____)

d. Are you choosing the nine athletes with or without replacement?

Short answer

a. The random variable X is the number of patriots picked.

b. The values of X is $0,1,2,3,4$

c. The distribution of X is $X~B(4,8,9)$

d. The nine athletes are chosen without replacement.

Step-by-step solution

Content Introduction

The hypergeometric distribution is utilized when you want to know how likely it is to get a certain number of successes from a given sample size without replacement.

Explanation (part a)

Random variable in simple terms generally refers to variables whose values are unknown, therefore, in this case X is the number of patriots picked.

Explanation (part b)

Make the list of values that you want to use X may take on.

As we can see there is an upper bound for the situation at hand so,

$X=0,1,2,3,4$

Explanation (part c)

The random variable X refers to the number of trials before the first success. Each trial is independent of others and has similar probability of success. This implies that random variable X follows Hypergeometric Distribution.

$X~B(4,8,9)$

where,

role="math" localid="1649171849421" $r=4,b=8,n=9$

Explanation (part d)

The nine athletes are chosen without replacement because mostly the chosen athletes are from Boston Celtics .