Question

1. A survey reflects the views of 1,000 undergraduate students on the new building on campus. The table below shows their points (like, dislike or not care) and which year they are. (write out all the probabilities in formal notion)

	Like	Dislike	Not care	Total
Freshman	100	98	22	220
Sophomore	152	85	23	260
Junior	80	178	12	270
Senior	143	76	31	250
Total	475	437	88	1000

1. Form the proportions table.
2. Are disliking the building and being junior mutually exclusive?
3. What is the probability that a randomly chosen student who is a freshman or like the building?
4. What is the probability that a randomly chosen student who is a sophomore given that he dislikes the building?
5. What is the probability that a randomly chosen student doesn’t care about the building given that he is a senior?
6. What is the probability that a randomly chosen student who is a sophomore or junior given that he likes the building?
7. Use general multiplication rule to determine the probability that a student both is a senior and don’t care about the building.
8. Does it appear that whether or not a student likes the building is independent of their years? Explain your reasoning.

Answer 1

a) Proportions table:

	Like	Dislike	Not care	Total
Freshman	100 / 1000 = 0.100	98 / 1000 = 0.098	22 / 1000 = 0.022	220 / 1000 = 0.22
Sophomore	152 / 1000 = 0.152	85 / 1000 = 0.085	23 / 1000 = 0.023	260 / 1000 = 0.26
Junior	80 / 1000 = 0.080	178 / 1000 = 0.178	12 / 1000 = 0.012	270 / 1000 = 0.27
Senior	143 / 1000 = 0.143	76 / 1000 = 0.076	31 / 1000 = 0.031	250 / 1000 = 0.25
Total	475 / 1000 = 0.475	437 / 1000 = 0.437	88 / 1000 = 0.088	1

b) P(Junior and Dislike) = 178/1000 = 0.178

No, disliking the building and being junior is not mutually exclusive.

Mutually exclusive events are things that can't happen at the same time. But as these event are happening at the same time we cannot say that they are mutually exclusive.

c) P(Freshman or Like) = P(Freshman) + P(Like) - P(Freshman and Like)

= 220/1000 + 475/1000 - 100/1000 = 0.595

d) P(Sophomore | Dislike) = P(Sophomore and Dislike) / P(Dislike)

= (85/1000) / (437/1000) = 0.1945

e) P(Not care | Senior) = P(Not care and Senior) / P(Senior)

= (31/1000) / (250/1000) = 0.124

f) P(Sophomore or Junior | Like) = P((Sophomore or junior) and Like) / P(Like)

= ((152+80)/1000) / (475/1000) = 0.4884

g) P(Senior and Not care) = 31/1000 = 0.031