Chapter 5, Problem T5.11SPT

(a)

To determine

To calculate: The probability that the student eats regularly in the cafeteria and is not a ${{\displaystyle 10}}^{th}$ - grader.

Answer to Problem T5.11SPT

The probability that the student eats regularly in the cafeteria and is not a ${{\displaystyle 10}}^{th}$ grader is 0.3975.

Explanation of Solution

Given information: Below is the table of the division of students

	Grade
		9th	10th	11th	12th	Total
Eat in cafeteria	Yes	130	175	122	68	495
	No	18	34	88	170	310
	Total	148	209	210	238	805

Total number of students $=805$

Formula used:

P (a student eats regularly in the cafeteria $\cap$ not a 10^th grader) $=\frac{\text{Number of students who eats regulary and is a 9th grader or 11th grader or 12th grader}}{\text{Total number of students}}$

Calculation:

From the given information,

Number of students who eats regularly in the cafeteria and is a 9^th grader $=130$

Number of students who eats regularly in the cafeteria and is a 11^th grader $=122$

Number of students who eats regularly in the cafeteria and is a 12^th grader $=68$

So, P (a student eats regularly in the cafeteria $\cap$ not a 10^th grader) $=\frac{130+122+68}{805}=\frac{320}{805}$

  $=0.3975$

(b)

To determine

To calculate: The probability that a student is a 10^th grader given the students eats regularly in the cafeteria.

Answer to Problem T5.11SPT

The probability that a student is a 10^th grader given the students eats regularly in the cafeteria is 0.3535.

Explanation of Solution

Formula used:

P (a student is a 10^th grader $|$ eats regularly at a cafeteria)

  $=\frac{\text{P (a student is a 10th grader }\cap \text{ eats regularly at a cafeteria)}}{\text{P (eats regularly at a cafeteria) }}$

P (eats regularly at a cafeteria) $=\frac{\text{Number of students who eat regularly at cafeteria}}{\text{Total number of students}}$

P (a student is a 10^th grader $\cap$ eats regularly in the cafeteria) $=\frac{\text{Number of students who are 10th grader and eat regularly at cafeteria}}{\text{Total number of students}}$

Calculation:

P (a student is a 10^th grader $\cap$ eats regularly in the cafeteria) $=\frac{175}{805}$

P (eats regularly at a cafeteria) $=\frac{495}{805}$

P (a student is a 10^th grader | eats regularly at a cafeteria) $=\frac{\frac{175}{805}}{\frac{495}{805}}=0.3535$

(c)

To determine

To check: Whether the events “10^th grader” and “eats regularly in the cafeteria” are independent.

Answer to Problem T5.11SPT

The events “10^th grader” and “eats regularly in the cafeteria” are not independent.

Explanation of Solution

Formula used:

Condition for independence

  $\text{P (10th grader }\cap P\text{ eats regularly in the cafeteria) = P (10th grader) }\times \text{ P (eats regularly in the cafeteria)}$

Calculation:

Number of students in 10^th grade $=209$

Number of students who eat regularly in the cafeteria $=495$

Number of students who are in 10^th grade and eat regularly in the cafeteria $=175$

So, P (10^th grade) $=\frac{209}{805}$

P (eats regularly in the cafeteria) $=\frac{495}{805}$

P (10^th grader and eats regularly in the cafeteria) $=\frac{175}{805}$

Checking condition of independence

  $\frac{175}{805}\ne \frac{209}{805}\times \frac{495}{805}$