We have a bag that contains 2 red balls. 3 blue balls and 4 green balls. For our first experiment, we take a random ball from the bag.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

 

a) We have a bag that contains 2 red balls. 3 blue balls and 4 green balls. For our first experiment, we take a random ball from the bag. observing its colour, return the ball to
the bag, and then take a second ball at random, observing its colour.

I) Describe the sample space and event space for this experiment. You do not need to explicitly list every element of the event space.

ii) Let DG denote the event in which the first ball drawn is blue, and the second is green. Calculate P( DC;).

iii) Let J? denote the event in which either the first or second ball drawn is red. Calculate P(R). In a second experiment, we keep the same bag. but instead of returning the first ball to the bag, we do not replace it before drawing the second.

iv) Explain how the probability space for this new experiment differs to that for the first experiment.

v) Let I?. B1. G1 be the events in which. respectively, a red, blue or green ball is drawn first, and G2 the event in which a green ball is drawn second. Calculate
P(G2IR,), P(G2IB1) and P(c;21G1).

vi) Using your answer from the previous question, calculate P(G2).

vii) Using Bayes Theorem, calculate P( B1 IC2).

b) Consider an infinite sequence of independent coin flips where the probability of each being heads is p. We define the family of random variables Xk to be the number of
heads after k flips.

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