Problem 13 - Chemistry: An Atoms-Focused Approach

Question

You flip three coins, assigning the values +1 for heads and -1 for tails. Each outcome of the three flips constitutes a micro state. How many different micro states are possible from flipping the three coins? Which value or values for the sums in the micro states are most likely? Hint: The sequence HHT \((+1+1-1)\) is one possible outcome, or micro state. Note, however, that this outcome differs from THH \((-1+1+1)\), even though the two sequences sum to the same value.

Step-by-Step Solution

Answer

Answer: The most likely values for the sums in the micro states are +1 and -1, with +1 being slightly more likely than -1.

Step 1: Analyze the coin flipping problem

There are three coins being flipped, which means there are 2 possible outcomes (H or T) for each coin. With three coins, the total number of micro states can be found using the formula: Total Micro states = \(2^n\), where n is the number of coins.

Step 2: Calculate the total number of micro states

Using the formula from Step 1: Total Micro states = \(2^3 = 8\). There are 8 possible micro states for flipping three coins.

Step 3: List all micro states and calculate their sums

Now let's list all the 8 possible micro states and the sum of their assigned values: 1. HHH: \((+1+1+1) = +3\) 2. HHT: \((+1+1-1) = +1\) 3. HTH: \((+1-1+1) = +1\) 4. THH: \((-1+1+1) = +1\) 5. HTT: \((+1-1-1) = -1\) 6. THT: \((-1+1-1) = -1\) 7. TTH: \((-1-1+1) = -1\) 8. TTT: \((-1-1-1) = -3\)

Step 4: Analyze the sums and determine the most likely value(s)

From the calculated sums: \(+3, +1, +1, +1, -1, -1, -1, -3\), we can observe the following: - The sum +1 has the highest frequency (4 times) - The sum -1 also appears quite frequently (3 times) - The sums +3 and -3 are least likely (appearing only once each) Therefore, the most likely values for the sums in the micro states are +1 and -1. However, +1 is slightly more likely than -1.

Key Concepts

Microstates

In probability theory, microstates refer to the different ways an event can occur. Microstates are the specific outcomes that comprise the overall event. When we talk about flipping a coin, each unique combination of heads (H) and tails (T) you get constitutes a microstate.

For example, if you flip three coins and label heads as +1 and tails as -1, each permutation of these flips is a microstate.

• HHH translates to the microstate: \(+1 + 1 + 1 = +3\)
• HHT translates to the microstate: \(+1 + 1 - 1 = +1\)
• And so on.

It's important to understand that each combination is distinct, even if the sum of their values coincides with another microstate. The uniqueness of the microstate lies in its sequence, not just in its possible sum. The sequence HHT and THH are considered different microstates even though they both sum to +1. Microstates are crucial in probability because they form the basis of calculating probabilities by considering all possible outcomes.

Coin Flipping

Coin flipping is a simple yet classic example in probability theory, often used to explain core concepts because each flip is an independent event with two possible outcomes: heads (H) or tails (T).

When more than one coin is flipped, the number of potential combinations (and thus microstates) increases exponentially. This is calculated by raising the number of outcomes per flip (2 for H or T) to the power of the number of flips.

For example, flipping three coins can yield:

• 2 options for the first coin (H or T)
• 2 options for the second coin
• 2 options for the third coin

The total number of microstates is calculated as:
\(2 \times 2 \times 2 = 2^3 = 8\). So, there are 8 possible combinations (or microstates) when flipping three coins. Coin flipping serves as the perfect foundation for introducing fundamental principles of probability because it presents clear, manageable outcomes.

Probability Distribution

A probability distribution describes how the probabilities are distributed over the different possible outcomes of a random experiment. It's a way to represent all the outcomes in terms of their likelihood.

When dealing with coin flips and microstates, each sum of values from the microstates contributes to the probability distribution. For instance, if you flip three coins and observe the sums of the values assigned to outcome, you'll end up with a distribution like:

• +3 occurring once
• +1 occurring four times
• -1 occurring three times
• -3 occurring once

This frequency determines the probability of each sum occurring. The probabilities can be calculated by dividing the frequency of each sum by the total number of microstates (which is 8 in this example).

Thus, the probability of getting a +1 is:
\[ \text{Probability of } +1 = \frac{4}{8} = 0.5 \]
The probability of a -1 is:
\[ \text{Probability of } -1 = \frac{3}{8} = 0.375 \]
The other sums, +3 and -3, occur with a probability of:
\[ \text{Probability of } +3 = \text{Probability of } -3 = \frac{1}{8} = 0.125 \]
Understanding probability distribution allows you to see which outcomes are the most or least likely.