91Ó°ÊÓ

In probability theory, a sample space is a collection of all possible outcomes of a particular random experiment. In the case of coin flipping, the experiment involves tossing a coin, where the outcomes are either heads (H) or tails (T). For a sequence of coin tosses, the sample space expands exponentially with each additional flip.

For 91 flips, each flip can result in either an H or a T, leading to a very large sample space. The number of all possible outcomes in this scenario is given by raising the number of outcomes per flip, 2, to the power of 91:

• Total number of outcomes = \(2^{91}\)

This immense number represents every conceivable sequence of heads and tails across 91 coin flips. Since all outcomes are equally likely, each sequence, like 91 heads in a row, is just one specific point in this enormous sample space.

The binomial distribution is a fundamental concept in probability that models the number of successes in a fixed number of independent trials, each with the same probability of success. For a coin flip, the trial is the flip itself, and the success can be defined as getting a head.

• Total number of trials: 91 (since Guildenstern flips the coin 91 times)
• Probability of success on each trial: \(0.5\) (for a fair coin, heads or tails each have a probability of \(\frac{1}{2}\))

The probability mass function of a binomial distribution helps us calculate the likelihood of getting a specific number of heads in this series of flips. For example, the probability of obtaining exactly 91 heads (i.e., all flips are heads) is calculated as:

• \[ \text{Probability} = \left(\frac{1}{2}\right)^{91} = \frac{1}{2^{91}} \]

This demonstrates the profound impact of repeated trials on probability — making such a streak of heads remarkably unlikely.

The complement rule is a useful tool in probability that remains especially handy when dealing directly with the probability of a complex event is cumbersome. It states that the probability of the complement of an event is equal to one minus the probability of the event itself.

• If \(A\) is an event, then \( P(A^c) = 1 - P(A) \)

In the problem scenario, we are asked to find the probability of getting at least one tail in 91 flips. Rather than calculating this directly, we use the complement rule:

• The event of 91 heads in a row has a probability of \( \frac{1}{2^{91}} \)
• The complement event, which includes at least one tail, thus has a probability of:
  \[ P(\text{at least one tail}) = 1 - \frac{1}{2^{91}} \]

Using the complement is often simpler and less prone to computational errors when managing probabilities in large sample spaces, especially with many trials like these coin flips.