Question

15.35% of the kids in AMS are Psychology majors. (PS) 30% of the students in AMS are Business majors. (B) 5% are both. Use a Venn diagram to help you answer the following: a. What is the probability that an AMS student selected at random is a Psychology major but not a business major? b. What is the probability that the student majors in neither Psychology nor Business? c. Are the events Psychology Major and Business Major independent or dependent? Explain mathematically how you arrived at your conclusion. d. Are the events P and B mutually exclusive or compatible? e. Find P(B/P) f. Find P(B or P)

Answer 1

The probability that the AMS is the Psychology majors: P(P) =0.35

The probability that the AMS is the Business majors: P(B) =0.30

$P(P\cap B)=0.05$

a) The probability that an AMS student selected at random is a Psychology major but not business major = $P(P)-P(P\cap B)=0.35-0.05=0.30$

b) The probability the student major is neither Psychology nor Business = $P(P'\cap B')=1-P(P\cup B)=1-0.35-0.30+0.05=0.40$

c) Two events are independent if the result of the second event is not affected by the result of the first event. If A and B are independent events, the probability of both events occurring is the product of the probabilities of the individual events.

   P(A and B)=P(A)⋅P(B)

Hence P and B are not independent but dependent.

$P(P\cap B)\neq P(P)P(B)$

d) Two events are mutually exclusive if only one can happen at a time.

i.e $P(P\cap B)=0$

Hence P and B are not mutually exclusive i.e compatible events as `

$P(P\cap B)\neq P(P)P(B)$

e) $P(B\mid P)=\frac{P(P\cap B)}{P(P)}=\frac{0.05}{0.35}=0.1429$

f) $P(B\cup P)=P(P)+P(B)-P(P\cap B)=0.35+0.30-0.05=0.60$