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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.

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.

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,

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

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

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

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

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