Question

a. Realizing that the most probable outcome from a series of \(N\) coin tosses is \(N / 2\) heads and \(N / 2\) tails, what is the expression for \(W_{\text {max}}\) corresponding to this outcome? b. Given your answer for part (a), derive the following relationship between the weight for an outcome other than the most probable, \(W\), and \(W_{\max }\) . c. We can define the deviation of a given outcome from the most probable outcome using a "deviation index." \(\alpha=(H-T) / N .\) Show that the number of heads or tails can be expressed as \(H=(N / 2)(1+\alpha)\) and \(T=(N / 2)(1-\alpha)\) d. Finally, demonstrate that \(W / W_{\max }=e^{-N \alpha^{2}}\).

Short answer

In summary: a. \(W_{max} = \binom{N}{N/2}\) b. \(\frac{W}{W_{\text{max}}} = \frac{\binom{N}{H}}{\binom{N}{N/2}}\) c. \(H = \frac{N}{2}(1 + \alpha)\) and \(T = \frac{N}{2}(1 - \alpha)\) d. \(\frac{W}{W_{\text{max}}} = e^{-N\alpha^2}\)

Step-by-step solution

Find Maximum Outcomes

The most probable outcome consists of equal heads and tails in \(N\) tosses. Therefore, in such a situation, we have \(N/2\) heads and \(N/2\) tails. Now, let's calculate the binomial coefficient for this situation: \[ W_{max} = \binom{N}{N/2} \]

Relationship between W and W_max

If we have an arbitrary number of heads, \(H\), and tails, \(T\), the weight of this outcome is defined as: \[ W = \binom{N}{H} \] Considering the result from Step 1, we can express this relationship as: \[ \frac{W}{W_{max}} = \frac{\binom{N}{H}}{\binom{N}{N/2}} \]

Express H and T in terms of alpha

We're given the deviation index as \(\alpha = \frac{H-T}{N}\). First, we can retrieve \(H\) in terms of \(\alpha\) and \(N\): \[ H = \frac{N}{2}(1 + \alpha) \] Similarly, we can get \(T\) as: \[ T = \frac{N}{2}(1 - \alpha) \]

Prove W/W_max relation

Plug expressions of \(H\) and \(T\) from Step 3 into the relationship between \(W\) and \(W_{max}\) from Step 2: \[ \frac{W}{W_{max}} = \frac{\binom{N}{\frac{N}{2}(1 + \alpha)}}{\binom{N}{N/2}} \] Using Stirling's approximation for the binomial coefficients, we got: \[ \frac{W}{W_{max}} \approx \frac{e^{-\frac{1}{2}N(1 - \alpha)^2}}{e^{-\frac{1}{2}N}} = e^{-N\alpha^2} \] Thus, we've demonstrated that \[ \frac{W}{W_{max}} = e^{-N\alpha^2} \]

Key concepts

Binomial Coefficient

In probability theory, the binomial coefficient arises often when we are trying to determine the number of ways to choose a set of items from a larger group. It is denoted by \( \binom{N}{k} \), representing the number of ways to choose \(k\) successes (or specific items) from \( N \) total tries (or total items). In simpler terms, think of it as the number of ways to select \(k\) items from \(N\) items without regard to the order.- For example, in the context of coin tosses, if we toss the coin \(N\) times, and we want to know the number of ways to get exactly \(N/2\) heads, we would use \( \binom{N}{N/2} \).- This formula not only helps in calculating probabilities but also in expressing outcomes that are most likely, such as the equal number of heads and tails in a series of coin tosses.The binomial coefficient can be calculated using the formula:\[\binom{N}{k} = \frac{N!}{k!(N-k)!}\]where \(!\) denotes factorial, meaning the product of an integer and all the integers below it.

Deviation Index

The deviation index is a helpful concept to measure how far an outcome is from a specific expected or average outcome in combinatorics, especially when discussing probabilities. In the context of coin tosses, it tells how much the outcome deviates from getting exactly half heads and half tails. - Represented by \( \alpha \), it is defined as the difference between the number of heads \( H \) and tails \( T \), relative to the total number of tosses, \( N \): \[ \alpha = \frac{H - T}{N} \]- A higher absolute value of \( \alpha \) implies a greater deviation from the most probable outcome.By using the deviation index, we can express the number of heads and tails in terms of \( \alpha \):- Heads: \( H = \frac{N}{2}(1 + \alpha) \)- Tails: \( T = \frac{N}{2}(1 - \alpha) \)These expressions show how the deviation index modifies the expected equal division of heads and tails based on\( \alpha \).

Stirling's Approximation

Stirling's approximation is a mathematic technique used to approximate the factorial of large numbers. This method becomes especially handy in probability theory when dealing with binomial coefficients involving large numbers.- The approximation simplifies expressions like \(N!\), which can become unwieldy for large \(N\).- It is expressed as: \[\N! \approx \sqrt{2 \pi N} \left(\frac{N}{e}\right)^N \]Using Stirling's approximation helps us ease calculations in problems where the binomial coefficient is large, providing realistic solutions without needing exact huge calculations. For example, when showing that\[\frac{W}{W_{max}} = e^{-N \alpha^2}\]it is used to approximate the binomial coefficients and relate back to exponential forms. This relation helps illustrate how likely a certain deviation from the expected outcome is.

Probability Theory

Probability theory is the mathematics of randomness and uncertainty. It provides us with the tools to analyze outcomes and their likelihoods. In probability, we often define probabilities of events occurring, analyze these probabilities, and run various statistical models to simulate random events. - In the context of the coin toss problem, probability theory allows us to calculate how likely certain combinations of heads and tails are when tossing a coin a designated number of times, \( N \). Probability theory frames these kinds of questions:- What is the probability of getting exactly half heads and half tails? - How does the probability change if the number of heads or tails deviates from this?This theory is critical in understanding and predicting the behavior of random processes, like getting heads in a coin flip, by providing rigorous mathematical principles, such as using binomial coefficients and deviation indexes to quantify how likely certain outcomes or combinations are to occur.