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Probability
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A total of 600 alumni, who graduated from Bachelor’s and Master’s programs were polled with the following information resulting

INSTRUCTIONS TO CANDIDATES
ANSWER ALL QUESTIONS

Problem 1 (22 points)

A total of 600 alumni, who graduated from Bachelor’s and Master’s programs were polled with the following information resulting:

 

Master

Bachelor

1st class honours

2nd class honours

3rd class honours

1st class honours

110

100

40

2nd class honours

200

121

45

3rd class honours

50

74

60

 

(a)         What is the probability that the student graduated from a Master’s program with 3rd class honours? (6 points)

(b)         What is the conditional probability that the student graduated from a Master’s program with 1st honour class given that he graduated from a Bachelor’s program with 1st class honour? (8 points)

(c)          What is the conditional probability that the student graduated from a Master’s program with 2nd class honour or higher given that he graduated from a Bachelor’s program with 3rd class honour? (8 points)

 

Problem 2 (20 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)

 

Problem 3 (22 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 6-faced 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)

 

Problem 5 (20 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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