Discuss the following:
Q1: An experiment consists of tossing an ordinary coin five times. Recall that an experiment can produce one or more sample spaces depending on what we consider our outcome to be. Consider the following 4 sets:
(a) S = {zero heads, one head, two heads, four heads, five heads}
(b) S = {no heads, no tails, at least one head}
(c) S = {more heads than tails, fewer heads than tails}
(d) S = {no heads, no tails, at least one head and one tail}
For each of the four sets, state whether or not it is a valid sample space for the experiment. If it is not a valid sample space, explain which of the two assumptions (mutually exclusive outcomes or exhaustive outcomes) has been violated.
Q2: What, if anything, is wrong with the following statements?
(a) When flipping a biased (unbalanced) coin and observing the side the coin lands on, our sample space is S = {H, T} and the probability of getting a head is 0.4 and the probability of getting a tail is 0.7.
(b) The probability that a programmer will have no errors in their software is 0.15 and the probability that they will have no more than 1 error in their software is 0.10.
(c) If I flip a fair coin and then roll a fair die, the probability I get a head followed by 5 dots is 1/2+1/6 = 2/3.