Question

Question 1 [12 + 4 =16 marks]

A. Let A and B be two events such that P( A)  0.6 , P(B)  0.4 and P( A  B)  0.10.

1. Calculate P( A  B).

2. Calculate P( A | B).

iii. Are events A and B independent? Justify your answer.
iv. Are events A and B mutually exclusive events? Justify your answer.

(2 + 2 + 3 + 3 = 10 marks)

B. A box contains 20 DVDs, 4 of which are defective.

i. If two DVDs are selected at random (without replacement) from this box, what is the probability that both are defective?

ii. If two DVDs are selected at random (with replacement) from this box, what is the probability that both are defective?

(3 + 3 = 6 marks)

Question 2 [ 6 + 8 + 4 = 18 marks]

A. Four factories are to be randomly selected from a list of eleven factories for a full health and safety inspection

i. Calculate the number of ways that four factories can be selected.
ii. Two of the factories have the same owner. Calculate the probability that both of these

factories with the same owner will be included in the inspection.

(3 +3=6marks)

B. The random variable X has the following probability distribution:

X	1	2	3
P(X)	k	3k	6k

i. Show that k. must equal 0.1
ii. Calculate the mean of X
iii. Calculate the standard deviation of X .

(2 +3+3=8marks)

C. Joe is playing a game of chance at the hibiscus festival, costing $1 for each game. In the game two fair dice are rolled and the sum of the numbers that turned up is found. If the sum is seven, then Joe wins $5. Otherwise loses his money. Joe play the game 15 times. Find his expected profit or lose.

(4 marks)

Question 3 [4 + 12 = 16 marks]

1. The basketball player has a 75% chance of a successful shot. The shots are assumed to be independent of each other. Find the probability that 3 out of the next 4 shots are successful.

   (4 marks)

2. The weight of potatoes produced by one particular farm are found to be approximately normally distributed, with the mean weight of 147 g and a standard deviation of 23g.

i. What proportion of the potatoes produced by this farm will weigh less than 160 g?
ii. Of all the potatoes produced on the farm, 5% are considered too light. What is the

maximum weight that will be considered too light?

(3 + 5 = 8 marks)

C. The amounts of electricity bills for all households in a city have a skewed probability distribution with mean of $140 and a standard deviation of $30. Find the probability that the mean amount of electric bills for a random sample of 75 households selected from the city will be within $6 of the population mean.

(4 marks)

Answer 1