Question

Suppose an undergraduate-admissions committee rated 400 applicants and randomly chose 12 from those in the top 15 %.

(i) Compute the probability that a person will be admitted given that he/she has the highest faculty rating among the 400 students.

(ii) Compute the probability that a person will be admitted given that he/she has the lowest faculty rating.

(b) In a production line, suppose 3 % of the products have Type A defects and 2 % of products have Type B defects. It is also known that 0.4 % of products have both types of defects. Given that a product is known to have Type A defect. Compute the probability that it has Type B defect.

(c) Dolomite is a common rock-forming mineral and the primary component of the sedimentary rock. During mining operations, dolomite is often mixed up with shale, which is another fine-grained sedimentary rock. Miners can make use of the radioactivity features of rock to help them distinguish between shale rock zone and dolomite rock zone. Based on certain guidelines and standards, if the gamma ray reading of a rock zone is less than 70 API units, the area is considered to be abundant in dolomite, and hence can be mined. On the other hand, if the gamma ray reading of a rock zone exceeds 70 API units, then the area is considered to be mostly shale and therefore will not be mined. In an exploratory research study, a random set of 750 sample data is collected from a rock quarry. It is found that 480 of the samples are dolomite and 270 of the samples are shale. Of the 480 dolomite samples, 50 of them had gamma rays readings greater than 70. As for the 270 share samples, 255 of them had gamma ray readings greater than 70. Suppose a gamma ray reading greater than 70 is obtained at a particular depth of the rock quarry.

Compute the probability that the area should be mined.

(d) The manager of a self-service carwash station found that customers take an average of 8 minutes to wash and dry their cars. Assuming that the self-service times can be modelled by exponential distribution, compute the probability that a customer will require more than 11 minutes to complete the job.

(e) In a manufacturing process where glass products are made, bubbles do occur. When this happens, the quality of the products will be affected. Suppose based on past records, on average, 1 in every 1000 of these glass products produced has one or more bubbles.

(i) Apply a suitable exact model to compute the probability that a random sample of 5000 glass products will result in fewer than 4 products with bubbles.

(ii) Apply an approximate model to compute the probability that a random sample of 5000 glass products will result in fewer than 4 products with bubbles.

(iii) Comment on the results. (1 mark)

Answer 1

Question 1) Suppose an undergraduate-admissions committee rated 400 applicants and randomly chose 12 from those in the top 15 %.

(i) Compute the probability that a person will be admitted given that he/she has the highest faculty rating among the 400 students.

Answer)

Probability = favorable/total

Total = 15% of 400 = 60

Favorable = who got admitted = 12

Probability = 12/60 = 0.2

(ii) Compute the probability that a person will be admitted given that he/she has the lowest faculty rating.

Answer)

Total = lowest faculty rating = 400-60 = 340

Favorable = got admitted = 0

Probability = 0/340 = 0

(b) In a production line, suppose 3 % of the products have Type A defects and 2 % of products have Type B defects. It is also known that 0.4 % of products have both types of defects. Given that a product is known to have Type A defect. Compute the probability that it has Type B defect.

Answer)

Lets say we have a sample size of 100

3% have type A defects = 3% of 100 = 3

2% have type B defects = 2% of 100 = 2

0.4% have both of these defects = 0.4% of 100 = 0.4

Total = type A defect = 3

Favorable = type B defect = 0.4

Probability = 0.4/3 = 0.1333