Consider a fair coin which when tossed results in either heads (H) or tails (T). If the coin is tossed TWO times

Question

(a) List all possible outcomes. (Order matters here. So, HT and TH are not the same outcome.)

(b) Write the sample space.

(c) List ALL possible events and compute the probability of each event, assuming that the probability of each possible outcome from part (a) is equal. (Keep in mind that there should be many more events than outcomes and not all events will have the same probability.

Answer 1

HH

HT

TH

TT

four possible outcomes, which comprise the sample space. 4 outcomes comprising 2 events each, for a seeming total 4x2 = 8 events.

probability of each outcome is the same, 1/4 or 0.25, possibility of each single event is 1/2 = 0.5

Only possibly different probability is the "Sleeping Beauty" controversy or paradox, or the "Monty Hall" problem where Sleeping Beauty evokes "thirders" and "halfers" disagreeing on whether to change your probability that a coin is heads or tails or which of 3 doors to choose, when there seemingly is no new information to make you change. Bayes Theorem gives a different result from frequentist traditional statistics. If you can somehow work that paradox into 2 coin flips, then you'd have a very paradoxical result.

Bayes Theorem says if you pick door A, and Monty opens door B, then it makes no difference if you switch to door C, yet computer programs with random numbers suggests you should switch to door C, as odds have changed from 1/2 to 1/3 for door A. But apply Bayes Theorem, a deductive approach and the 1/2 remains 1/2. Deduction or Induction? same with the coin flip in Sleeping Beauty. She wakes up and is asked if she wants to change her bet on the coin that was already flipped. She says yes, according to a computer program, but not according to Bayes Theorem. Supposedly the math profession sides with switching doors in Monty Hall's problem, but only a majority of mathematicians say Sleeping Beauty should switch, with a strong minority dissent.