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.

a. What is the interpretation of conditional probability in terms of new information? b. Is the conditional probability of A given B always the same number as the conditional probability of B given A? c. How can you find the conditional probability from the probabilities of two events and the probability of their intersection? d. What are the conditional probabilities for two independent events? e. Is the conditional probability of A given B a probability about A or a probability about B?ed. Outcome Probability Conditional probability = called for Called all 3 0.15 called Jury duty years Conditional probability for jury 0.15 not called Called the 1st and duty (0.15)10.85) - Conditional for Jury 2nd years but not 0.019125 called probability = duty the 3rd 0.85 for jury Conditional Probability probability called for Called the ist and 0.15 duty not 0.15 jury duty 3rd years but not (0.15)(0.85)(0,15)= called the 2nd 0.019125 Conditional probability for jury not called Called the 1st 0-85 duty Conditional for jury year but not the probability = duty 2nd and 3rd 0.85 Conditional called for Not called the 1st probability = 0.15 year but called the (0.85)10.15)= called jury duty 0,019125 2nd and 3rd years Conditional for jury probability 0.15 duty not called Not called the ist not Conditional for jury or 3rd year but (0.85)(0.15)(0.85)- Probability probability = 0. 108375 0.85. called 0.85 duty called the 2nd year for jury Conditional duty not probability called for Not called the ist or (0.85) (0.15)- 0.15 called jury duty 2nd year but called 0.108375 Conditional the 3rd year probability = for jury 0.85 not called P(chosen at least duty Conditional for jury Not called any (0.8513 - 0.614125 probability = duty of the 3 years 1 year ) = 0.85QUESTION 12 Select the comect statement concerning the Law of Total Probability and conditional probability. The Law of Total Probability defines unconditional probability of an event P(E) using known conditional probabilities P(BB) given that independent events Bk have occurred, each Bk with a unknown probability itself. Conditional probability P(E F) defines the probability of an event E given that an event F has occurred. Ob. The Law of Total Probability defines unconditional probability of an event P(E) using known conditional probabilities P(E B) given that mutually exclusive events B& have occurred, each BA with a known probability itself. Conditional probability P(E) ) defines the probability of an event F given that an event E has occurred. The Law of Total Probability defines unconditional probability of an event P(E) using known conditional probabilities P(E)By) given that independent events BA have occurred, each BK with a known probability itself. Conditional probability P(E F) defines the probability of an event F given that an event E has occurred. The Law of Total Probability defines unconditional probability of an event P(2) using known conditional probabilities P(FB) given that mutually exclusive events By have occurred, each By with a known probability itself. Conditional probability P(EA) defines the probability of an event E given that an event F has occurred.9) The Logit model allows for nonconstant marginal effects. This means that a. The conditional probability of y; = 1 given x, is nonlinear in xi. b. The conditional probability of y; = 1 given x; is nonlinear in the model coefficients. c. The derivative with respect to x; of the conditional probability of y; = 1 given x; is nonlinear in xi. d. The derivative with respect to x; of the conditional probability of y; = 1 given x; is nonlinear in the model coefficients