Question

Problem numbers in red indicate that the solution to the problem is given in the Student's Solutions Manual.a. What is the possible number of microstates associated with tossing a coin \(N\) times and having it come up \(H\) times heads and \(T\) times tails? b. For a series of 1000 tosses, what is the total number of microstates associated with \(50 \%\) heads and \(50 \%\) tails? c. How much less probable is the outcome that the coin will land \(40 \%\) heads and \(60 \%\) tails?

Short answer

a. The number of microstates associated with tossing a coin N times and having it come up H times heads and T times tails can be calculated using the Combination formula: \( \binom{N}{H} = \frac{N!}{H!(N-H)!} \). b. For a series of 1000 tosses with 50% heads and 50% tails, the total number of microstates is \( \binom{1000}{500} = \frac{1000!}{500!500!} \). c. The probability of 40% heads and 60% tails compared to 50% heads and 50% tails can be calculated by comparing the combinations: \( \frac{\binom{1000}{400}}{\binom{1000}{500}} \). Calculate the numerical values to find out how much less probable this outcome is.

Step-by-step solution

a. Calculate the number of microstates associated with an N-times coin toss

To find the number of possible microstates when tossing a coin N times and getting H times heads and T times tails, we will use the Combination formula as follows: \[ \binom{N}{H} = \frac{N!}{H!(N - H)!} \] Here, N represents the total number of tosses, and H represents the number of heads. Since we know that H + T = N, the position of tails is already determined by the number of heads, so there is no need to consider T in the formula.

b. Number of microstates associated with 50% heads and 50% tails in 1000 tosses

Given 1000 tosses, 50% heads and 50% tails mean 500 heads and 500 tails. We'll calculate the combination using the total number of tosses (N = 1000) and the number of heads (H = 500): \[ \binom{1000}{500} = \frac{1000!}{500!500!} \] To find the numerical result, a calculator or computer program can be used.

c. The probability of 40% heads and 60% tails compared to 50% heads and 50% tails

We will calculate the probabilities for 40% heads and 60% tails and compare it to the probability found in part b. For 40% heads and 60% tails, we will have 400 heads and 600 tails. Using the combination: \[ \binom{1000}{400} = \frac{1000!}{400!600!} \] Now, we need to compare the probabilities: \[ \frac{\binom{1000}{400}}{\binom{1000}{500}} \] By calculating the numerical values using a calculator and/or a computer program, you can find out how much less probable the outcome of 40% heads and 60% tails is compared to the 50% heads and 50% tails scenario.

Key concepts

Understanding the Combination Formula

The combination formula is a crucial tool in probability, especially in problems involving the arrangement of items like in a coin toss experiment. It is expressed as:
\[\binom{N}{H} = \frac{N!}{H!(N - H)!}\] Here, \(N\) represents the total number of trials or events, while \(H\) represents the specific outcomes you are interested in, such as the number of heads in a series of coin tosses.

• \(N!\) ("N factorial") is the product of all positive integers up to \(N\).
• \(H!\) and \((N-H)!\) are similar factorials for heads and tails, respectively.

This formula helps determine the number of ways to choose a specific number of successes (\(H\) heads) from \(N\) trials. Don’t worry about calculating it by hand; factorials grow rapidly, and often a calculator or a computer program is used for practical computation.

Calculating Probability

Probability is all about finding out how likely an event is to happen. Mathematically, it is the number of favorable outcomes divided by the total number of possible outcomes. For instance, if you want to know the probability of getting a certain number of heads in a coin toss series, you determine:
\[P = \frac{\text{Number of favorable microstates}}{\text{Total number of microstates}}\] In step c of the exercise, you see this applied to compare the probability of an experiment resulting in 40% heads and 60% tails to 50% heads and 50% tails. This involves finding the number of microstates for each scenario and using these to calculate the probabilities and their ratio.
This gives a direct comparison, showing how much less likely one outcome is compared to another.
Understanding these concepts helps to simplify solving probability-based problems and makes predicting outcomes more intuitive.

Exploring the Coin Toss Experiment

A coin toss experiment is a classic example used in probability and statistics to understand random processes. When tossing a coin multiple times, every toss can result in either heads or tails, and each result has a probability of \(0.5\) if the coin is fair.
The beauty of this experiment is in the predictable nature of large numbers. As you increase the number of tosses, you can apply formulas like combinations to find the number of microstates associated with specific outcomes, like how often half the tosses are heads.
When we deal with a large number of tosses, like 1000, calculating the exact number of each outcome using the combination formula helps you visualize how varied the results can be. This experiment illustrates fundamental principles of probability and showcases the predictable yet random nature of such processes.