Question

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)

Answer 1

a) The sample space of the experiment is {(4,5), (4,6), (4,7), (4,8), (4,9), (5,6), (5,7), (5,8), (5,9), (6,7), (6,8), (6,9), (7,8), (7,9), (8,9)}.

b) i. There are 5 outcomes where there are 13 carrots in total planted over the two days: (4,9), (5,8), (6,7), (7,6), (9,4). Therefore, the probability of this event is 5/15 or 1/3.

ii. There are 7 outcomes where an odd number of carrots were planted on the second day: (4,5), (4,7), (5,7), (6,7), (7,5), (7,7), (9,7). Therefore, the probability of this event is 7/15.

c) The events (i) and (ii) are mutually exclusive because there are no outcomes where both events occur.

d) The events (i) and (ii) are not statistically independent because the outcome of event (ii) affects the outcome of event (i). For example, if an odd number of carrots were planted on the second day, it is impossible for there to be an even number of carrots planted over the two days, which is a requirement for event (i) to occur. Therefore, the probability of event (i) is affected by the occurrence of event (ii).

1.2 a) It is not possible that P(AB)=0.1 because the probability of the intersection of two events cannot be greater than the probability of either event occurring alone. In other words, P(AB) ≤ P(A) and P(AB) ≤ P(B).

b) The smallest possible value of P(AB) is 0 because the intersection of two events cannot have a negative probability.

c) The largest possible value of P(AB) is 0.7 because P(AB) cannot be greater than the probability of event B occurring alone.

1.3 a) (AUB)(AUB) = AUB (distributive property)

b) (AUB)(AUB)(AB) = AUB (AB = A∩B, so (AUB)(AUB)(AB) = AUB∩AUB∩B = AUB∩B =