Question

Use the following information to answer the next seven exercises: A ballet instructor is interested in knowing what percent of each year's class will continue on to the next, so that she can plan what classes to offer. Over the years, she has established the following probability distribution.

- Let$\mathrm{X}=$the number of years a student will study ballet with the teacher.

- Let $\mathrm{P}\left(\mathrm{x}\right)=$the probability that a student will study ballet $\mathrm{x}$years.

What does the column $"\mathrm{P}\left(\mathrm{x}\right)"$sum to and why?

Short answer

Sum of $\mathrm{P}\left(\mathrm{x}\right)$is $1$and is to be a probability of a students will study ballet.

Step-by-step solution

Table for no of students probabilty

Calculation of sum

For sum of $\mathrm{P}\left(\mathrm{x}\right)$:

From table,

$\mathrm{P}\left({\mathrm{x}}_1\right)=0.1$,$\mathrm{P}\left({\mathrm{x}}_2\right)=0.05$

$\mathrm{P}\left({\mathrm{x}}_3\right)=0.1$,$\mathrm{P}\left({\mathrm{x}}_4\right)=0.15$

$\mathrm{P}\left({\mathrm{x}}_5\right)=0.3$,localid="1649787457135" $\mathrm{P}\left({\mathrm{x}}_6\right)=0.2$,localid="1649787464973" $\mathrm{P}\left({\mathrm{x}}_7\right)=0.1$

So,

$\mathrm{P}\left(\mathrm{x}\right)=\sum _{\mathrm{i}=1}^7\mathrm{P}\left({\mathrm{x}}_{\mathrm{i}}\right)$

$=0.1+0.05+0.1+0.15+0.3+0.2+0.1$

$=1$

Sum of $\mathrm{P}\left(\mathrm{x}\right)$is probability of $\mathrm{x}$.