A local farmer plants a given number carrots on a certain number of days. We are looking at the number of carrots the farmer can plant over two days. Suppose that the famers must plant at least 4 carrots on the first day, no more than 9 carrots on the second day and farmer has to plant more carrots on the second day than the first day. a) Determine the sample space of the experiment. b) If each of the outcomes in (a) have equal probability of occurring find the probability of the following events: i. Event that there were 13 carrots in total planted over the two days. ii. Event that an odd number of carrots were planted on the second day. c) Are the events (i) and (ii) mutually exclusive? Motivate your answer! d) Are the events (i) and (ii) statistically independent? Motivate your answer! Question 1.2 [2, 2, 21 Suppose that we have two events A and B such that P(4)=0.8 and P(B)=0.7. a) Is it possible that P(AB)=0.1? Explain your answer. b) What is the smallest possible value of P(AB)? c) What is the largest possible value of P(AB)? Question 1.3 [2, 2, 21 Given the following three events A, B and C, find simpler expressions for the following: a) (AUB)(AUB) b) (AUB)(AUB)(AB) c) (AUB)(BUC) Question 1.4 [3.11 A fair coin is tossed three times a) What is the probability of obtaining two or more heads given that there was at least one head is obtained? b) What is the probability of at least one tail? Question 1.5 [4] If B is an event, with P(B)>0, show that the following is true P(AUC|B)=P(A/B)+P(C\B)~P(A^C\B)
