91Ó°ÊÓ

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 \(\mathrm{c}\) and \(\mathrm{d}\) are based.

Step by step solution

01

Understanding the Sample Space

The sample space for a sequence of 91 coin flips can be represented by all possible outcomes where each flip results in either heads (H) or tails (T). Each flip is independent and has 2 possible outcomes (H or T). Thus, for 91 flips, the sample space consists of sequences of H's and T's, each of length 91.

02

Calculating the Number of Outcomes

Since each of the 91 flips has 2 possible outcomes, the total number of outcomes in the sample space is \(2^{91}\). This means there are \(2^{91}\) possible sequences of heads and tails.

03

Identifying Specific Outcomes

Guildenstern's outcome is one where all 91 flips resulted in heads, represented as HHH...H (91 times H in sequence). The outcome where only the second flip is a tail is represented as HTT...T (starting with an H, then a T, followed by 89 more H's).

04

Calculating Probability of All Heads

The probability of getting a head in one flip is \(\frac{1}{2}\). For 91 consecutive heads, it is \(\left(\frac{1}{2}\right)^{91}\). This represents the probability of one specific outcome among the \(2^{91}\) possible outcomes.

05

Calculating Probability of At Least One Tail

The probability of getting at least one tail is the complement of getting all heads. Using the complement rule, it is calculated as \(1 - \left(\frac{1}{2}\right)^{91}\).

06

Explaining the Probability Model

The solutions for parts c and d are based on a uniform probability model where each outcome in the sample space is equally likely. The probability of each outcome happening is \(\frac{1}{2^{91}}\), assuming the coin is fair and each flip is independent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Space

When dealing with probability, the concept of sample space is crucial. The sample space is all the possible outcomes of an experiment. In the case of coin flips, each flip can result in heads (H) or tails (T).
For 91 coin flips, the sample space is the set of all sequences consisting of 91 outcomes, where each outcome is either H or T. Thus, each sequence, such as "HHHT...", represents a potential result of the series of flips.
Importantly, the size of the sample space is given by the number of possible outcomes for each individual coin flip (2 in this case, H or T), raised to the power of the number of flips (91). This results in the total number of outcomes being \(2^{91}\).

Coin Flips

Coin flipping is a simple probabilistic event that is often used to explore basic principles of probability. A single coin flip has two possible outcomes, heads (H) or tails (T), each with a probability of \(\frac{1}{2}\) if the coin is fair.
When flipping a coin multiple times, the outcomes of each flip are combined to form sequences, such as HHH, HTH, or TTT. In the scenario described, 91 flips of a fair coin result in a vast number of possible sequences, specifically \(2^{91}\).
Each flip is independent, meaning the outcome of one flip does not affect the others. This independence is key to calculating the probabilities of specific sequences.

Independent Events

Independent events are those where the occurrence of one event does not influence the outcome of another. In the context of coin flips, each flip is independent of the others.
This independence implies that the probability of getting heads or tails in any single flip remains constant at \(\frac{1}{2}\), regardless of previous outcomes. This property is fundamental when considering sequences of flips like the 91 in the exercise.
With independence, to find the probability of a specific sequence occurring, such as getting all heads, you multiply the probabilities of each individual event. For 91 flips resulting in heads, the probability is \(\left(\frac{1}{2}\right)^{91}\).

Complement Rule

The complement rule is useful when you need to find the probability of "at least one" event happening. Instead of calculating the probability directly, it's often easier to find the probability of the complement event happening and subtracting this from 1.
For example, to find the probability of getting at least one tail in 91 flips, you start by calculating the probability of getting no tails, which is the probability of all heads.
This is \(\left(\frac{1}{2}\right)^{91}\). The complement rule states that the probability of at least one tail is 1 minus the probability of getting all heads, resulting in \(1 - \left(\frac{1}{2}\right)^{91}\).
Utilizing the complement rule simplifies calculations, especially in large sample spaces.