Question

A group of Martial Arts students is planning on participating in an upcoming demonstration. Six are students of Tae Kwon Do; seven are students of Shotokan Karate. Suppose that eight students are randomly picked to be in the first demonstration. We are interested in the number of Shotokan Karate students in that first demonstration.

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. How many Shotokan Karate students do we expect to be in that first demonstration?

Short answer

a. The random variable X is the number of Shotokan Karate students in that first demonstration.

b. The values of X are$0,1,2,3,4,5,6,7$

c. The distribution of X is$X~Hypergeometric(7,6,8)$

d. The number of Shotokan Karate students do we expect to be in that first demonstration is$4.31$

Step-by-step solution

Content Introduction

A discrete probability distribution, the hypergeometric distribution. It's utilised 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 Shotokan Karate students in that first demonstration.

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,

$X=0,1,2,3,4,5,6,7$

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.

$X~Hypergeometric(7,6,8)$

where,

$r=7$is the number of students in Shotokan Karate

$b=6$the number of students in Tae Kwon do

$n=8$is the number of items in sample

Explanation (part d)

The expected number of Shotokan Karate students in the first demonstration is

$n=8,r=7,b=6$

$$\begin{gathered} E\left(X\right)=\frac{nr}{r+b} \\ E\left(X\right)=\frac{8\times 7}{7+6} \\ E\left(X\right)=4.31 \end{gathered}$$