QUESTION ONE

A triangular spinner has one red side, one blue side and one green side. The red side is weighted so that the spinner is four times more likely to land on the red side than on the blue side.

The green side is weighted so that the spinner is three times more likely to land on the green side than on the blue side.

a. Show that the probability that the spinner lands on the blue side is 18.

b. The spinner is spun 3 times. Find the probability that it lands on a different colored side each time.

c. The spinner is spun 136 times. Use a suitable approximation to find the probability that it lands on the blue side fewer than 20 times.

QUESTION TWO

In a group of 30 teenagers, 13 of the 18 males watch 'Kops are Kids' on television and 3 of the 12 females watch 'Kops are Kids'.

a. Find the probability that a person chosen at random from the group is either female or watches 'Kops are Kids' or both.

b. Showing your working, determine whether the events 'the person chosen is male' and 'the person chosen watches Kops are Kids' are independent or not.

QUESTION THREE

There are three sets of traffic lights on ???????'? journey to work. The independent probabilities that ??????? has to stop at the first, second and third set of lights are 0.4, 0.8 and 0.3 respectively.

a. Draw a tree diagram to show this information.

b. Find the probability that ??????? has to stop at each of the first two sets of lights but does not have to stop at the third set.

c. Find the probability that ??????? has to stop at exactly two of the three sets of lights.

d. Find the probability that ??????? has to stop at the first set of lights, given that she has to stop at exactly two sets of lights.

QUESTION FOUR

Rachel and Anna play each other at badminton. Each game results in either a win for Rachel or a win for Anna. The probability of Rachel winning the first game is 0.6. If Rachel wins a particular game, the probability of her winning the next game is 0.7, but if she loses, the probability of her winning the next game is 0.4. By using a tree diagram, or otherwise.

a. find the conditional probability that Rachel wins the first game, given that she loses the second.

b. find the probability that Rachel wins 2 games and loses 1 game out of the first three games they play.

QUESTION FIVE

Jason throws two fair dice, each with faces numbered 1 to 6. Event A is 'one of the numbers obtained is divisible by 3 and the other number is not divisible by 3'. Event B is 'the product of the two numbers obtained is even'.

a. Determine whether events A and B are independent, showing your working.

b. Are events A and B mutually exclusive? Justify your answer.

Which of the following about Central Limit Theorem is not correct? 0 The central limit theorem does not apply when n is small, eg n 00. vi) Hence or otherwise justify that X (1) is a consistent estimator of I9. b) Now suppose that X[1},X(2),X{3),X(4),X{5J are the order statistics of a random sample of size ve from this distribution. Let the observed value of X (1) be an]. The test rejects H0 : t9 = 2 and accepts H1 :6 5L 1 when either 3(1) 2 2 or 3(1)