
Master 

Bachelor 
1^{st} class honours 
2^{nd} class honours 
3^{rd} class honours 
1^{st} class honours 
110 
100 
40 
2^{nd} class honours 
200 
121 
45 
3^{rd} class honours 
50 
74 
60 
(a) What is the probability that the student graduated from a Master’s program with 3^{rd} class honours? (6 points)
(b) What is the conditional probability that the student graduated from a Master’s program with 1^{st} honour class given that he graduated from a Bachelor’s program with 1^{st} class honour? (8 points)
(c) What is the conditional probability that the student graduated from a Master’s program with 2^{nd} class honour or higher given that he graduated from a Bachelor’s program with 3^{rd} class honour? (8 points)
5 students are to be chosen from a group of 10 freshmen, 8 sophomores, and 12 juniors.
(a) What is the probability that at least two sophomores are chosen? (6 points)
(b) What is the probability that exactly one junior is chosen? (6 points)
(c) Given that no freshmen are chosen, what is the conditional probability that there are exactly three sophomores among the 5 chosen? (8 points)
For 6 coin tossed, compute the probabilities of the following events:
(a) All tosses are head. (4 points)
(b) The first toss is tail and the others are heads. (4 points)
(c) Exactly 4 heads. (6 points)
(d) The last three are tails. (4 points)
(e) There are at most 3 heads. (4 points)
Problem 4 (16 points)
Suppose we have six 6faced dice. Every th die has faces each with number “1” and the other faces with number “2” (). When one of the dice is randomly selected and rolled, it shows “1”. What is the conditional probability that
(a) it was the 2nd die? (8 points)
(b) it was the 6th die? (8 points)
The university has conducted a vaccination survey among students. 40% of students are Freshmen, 29% are Sophomore, 7% are Junior and 24% are Senior. 35% of Freshmen, 62% of Sophomores, 40% of Juniors and 58% of Seniors were vaccinated. A student is randomly chosen. Given that this student has been vaccinated, what is the probability that
(a) the student is a Freshman, (5 points)
(b) the student is a Junior, (5 points)
(c) the student is a Senior (5 points)
(d) what percent of students were vaccinated? (5 points)
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