Rosencrantz and Guildenstern In the opening scene of Tom Stoppard's play Rosencrantz and Guildenstern Are Dead, about two Elizabethan contemporaries of Hamlet, Guildenstern flips a coin 91 times and gets a head each time. Suppose the coin was balanced.

a. Specify the sample space for 91 coin flips, such that each outcome in the sample space is equally likely. How many outcomes are in the sample space?
b. Show Guildenstern's outcome for this sample space. Show the outcome in which only the second flip is a tail.
c. What's the probability of the event of getting a head 91 times in a row?
d. What's the probability of at least one tail, in the 91 flips?
e. State the probability model on which your solutions in parts c and d are based.

a. The sample space for 91 coin flips would consist of all possible sequences of heads (H) and tails (T) of length 91. Each outcome in the sample space would represent a specific sequence of heads and tails that could occur. There are 2^91 possible outcomes in the sample space.

b. Guildenstern's outcome for this sample space would be a sequence of 91 consecutive heads: HHH...HHH. The outcome in which only the second flip is a tail would be: HTHHH...HHH (with 89 more heads following).

c. The probability of getting a head 91 times in a row can be calculated by taking the probability of getting a head on a single flip and raising it to the power of 91. Since the coin is balanced, the probability of getting a head on a single flip is 1/2. Therefore, the probability of getting a head 91 times in a row is (1/2)^91.

d. The probability of getting at least one tail in the 91 flips can be calculated by subtracting the probability of getting all heads from 1. Since the probability of getting all heads is (1/2)^91, the probability of getting at least one tail is 1 - (1/2)^91.

e. The solutions in parts c and d are based on the assumption that the coin flips are independent and equally likely, which is the basis of the binomial probability model.

a. To specify the sample space for 91 coin flips, we need to consider all possible outcomes for each flip. Since the coin is balanced, there are two possible outcomes for each flip: heads (H) or tails (T).

Therefore, the sample space for 91 coin flips consists of all possible combinations of heads and tails for each flip. Each flip is independent, so we can multiply the number of outcomes for each flip to find the total number of outcomes in the sample space.

Since there are 2 possible outcomes for each flip, and there are 91 flips in total, the sample space consists of 2^91 or approximately 7.9228 x 10^27 outcomes.

b. Guildenstern's outcome for this sample space is obtaining a head (H) in all 91 flips. This outcome can be represented as HHH...HH (91 H's).

The outcome in which only the second flip is a tail (T) can be represented as HTTT...TT (90 T's).

c. The probability of getting a head 91 times in a row can be calculated by dividing the number of favorable outcomes (1) by the total number of possible outcomes (2^91).

Therefore, the probability is 1 / 2^91 or approximately 5.4259 x 10^-28.

d. The probability of getting at least one tail in the 91 flips can be calculated by subtracting the probability of getting all heads from 1.

Since we calculated in part c that the probability of getting all heads is approximately 5.4259 x 10^-28, the probability of getting at least one tail is 1 - 5.4259 x 10^-28, which is essentially equal to 1.

e. The solutions in parts c and d are based on the assumption that the coin flips are independent and each flip has an equal probability of resulting in heads or tails. This is known as the Bernoulli probability model, where each trial (coin flip) has two possible outcomes with equal probabilities.