1. Question Two [25 Marks]

a. Due to COVID-19 pandemic, the South African government is awarding students with bursary and daily free meals. Government had subjectively estimated that each student has a 70% probability of being approved to receive a bursary and a 67% probability of being approved to receive daily free meals if the applicant obtains bursary. If the application to receive a bursary was rejected, there is 50% probability of getting approval to receive daily free meals. Find the probability of an applicant getting approval to receive a bursary given that approval was already given for daily free meals. [15]

b. KZY Farms is located some 200 km from Durban. The farm rears cattle, chicken and pigs. There have recently been an outbreak of a bacteria disease within the piggery section of the farm. It has been estimated that the probability of finding a pig which would test positive to the bacterial infection is 30%. Calculate the number of pigs that would have to be tested if a laboratory technologist would be 99% sure of getting at least one positive result. [10]

1. Question Three [25 Marks] [25 Marks]

a. The random variable Y is B (3, 0.7). Construct the probability distribution for X and find the mean and standard deviation. [15]

b. Suppose X follows a Poisson distribution with standard deviation of 2.5. Find P(X=5). [10]

Question Four [25 Marks]

a. Out of the eggs that were laid by birds that were fed with bambara groundnut fortified feed, 20% are extra large. If a random sample of 400 eggs is taken, find the probability that less than 100 eggs will be extra-large. [15]

b. A sample of 40 farmers comprises of 25 adopters and 15 non-adopters of organic fertilizer. Also 15 the adopters grow cowpea while 7 of the non-adopters grow rice. Determine the probability that the first person interviewed was either a non-adopter or someone that grow cowpea. [10]

Question Five [25 Marks]

a. A bag contains 100 fruits of which there are 60 oranges and 40 grapes. What is the probability that trying 6 times and replacing the fruits before drawing the next one, we shall get:

i. Exactly 5 oranges.

ii. At least 3 oranges. [10]

b. A weight reducing program that includes a strict diet and exercise claims on its online advertisement that it can help an average overweight person lose 10 kgs in three months. Following the program's method, a group of twelve overweight persons have lost 8.1, 5.7, 11.6, 12.9, 3.8, 5.9, 7.8, 9.1, 7.0, 8.2, 9.3 and 8.0 kgs in three months. Test at 5% level of significance whether the program's advertisement is overstating the reality.

[15]