Answer to Question #126784 in Statistics and Probability for jaya

Question

04) a) In a campus restaurant it was found that 35% of all customers order vegetarian meals and that
50% of all customers are students. Further, 25% of all customers who are students order
vegetarian meals.
I. What is the probability that a randomly chosen customer both is a student and orders a
vegetarian meal?
II. If a randomly chosen customer orders a vegetarian meal, what is the probability that the
customer is a student?
III. What is the probability that a randomly chosen customer both does not order a
vegetarian meal and is not a student?
IV. Are the events “customer orders a vegetarian meal” and “customer is a student”
independent?
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V. Are the events “customer orders a vegetarian meal” and “customer is a student”
mutually exclusive?

Answer 1

Let $V$ be the event of customers ordering vegetarian meals and $S$ be the event of customers being students.

$P(V)=0.35, P(S)=0.5, P(V|S)=0.25$

I. The probability that a randomly chosen customer both is a student and orders a

vegetarian meal

$$P(V\cap S)=P(S)P(V|S)=0.5\cdot0.25=0.125$$

II. If a randomly chosen customer orders a vegetarian meal, what is the probability that the customer is a student

$$P(S|V)={P(V\cap S)\over P(V)}={0.125\over 0.35}={5\over 14}\approx0.357$$

III. What is the probability that a randomly chosen customer both does not order a

vegetarian meal and is not a student?

From De Morgan's Laws

$$P(V^c\cap S^c)=P(\overline{V\cup S})$$

Then

$$P(V^c\cap S^c)=P(\overline{V\cup S})=1-P(V\cup S)=$$

$$=1-(P(V)+P(S)-P(V\cap S))=$$

$$=1-(0.35+0.5-0.125)=0.275$$

IV. Are the events “customer orders a vegetarian meal” and “customer is a student”

independent? 

$$P(V\cap S)=0.125\not=0.175=0.35\cdot 0.5=P(V)P(S)$$

Therefore the events “customer orders a vegetarian meal” and “customer is a student” are not independent.

V. Are the events “customer orders a vegetarian meal” and “customer is a student”

mutually exclusive?

$$P(V\cap S)=0.125\not=0$$

Therefore the events “customer orders a vegetarian meal” and “customer is a student” are not mutually exclusive.