Problem 84 - Chemistry The Science in Context

Question

You flip three coins, assigning the values +1 for heads and -1 for tails. Each outcome of the three flips constitutes a microstate. How many different microstates are possible from flipping the three coins? Which value or values for the sums in the microstates are most likely? (Hint: The sequence HHT \([+1+1-1]\) is one possible outcome, or microstate. 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 of the sums from flipping three coins are 1 and -1.

Step 1: Calculate the number of possible microstates

There are three coins and each coin has two possible outcomes (heads or tails). Therefore, the total number of possible microstates can be obtained by multiplying the outcomes of each coin: \(2 \times 2 \times 2 = 8\) microstates.

Step 2: List all the microstates and their sums

Now list all the possible microstates by flipping each coin and assign +1 for heads and -1 for tails: 1. HHH (Heads, Heads, Heads): \(+1 +1 +1 = 3\) 2. HHT (Heads, Heads, Tails): \(+1 +1 -1 = 1\) 3. HTH (Heads, Tails, Heads): \(+1 -1 +1 = 1\) 4. THH (Tails, Heads, Heads): \(-1 +1 +1 = 1\) 5. HTT (Heads, Tails, Tails): \(+1 -1 -1 = -1\) 6. THT (Tails, Heads, Tails): \(-1 +1 -1 = -1\) 7. TTH (Tails, Tails, Heads): \(-1 -1 +1 = -1\) 8. TTT (Tails, Tails, Tails): \(-1 -1 -1 = -3\)

Step 3: Determine the most likely value or values of the sums

Upon analyzing the sums of each microstate, we can see that the sums are distributed as follows: - One microstate with a sum of 3 - Three microstates with a sum of 1 - Three microstates with a sum of -1 - One microstate with a sum of -3 The most likely values for the sums are the ones which have the highest frequency. In our case, the two most likely values are \(1\) and \(-1\), since they each occur three times out of the eight possible microstates.

Key Concepts

Understanding Microstates

In the context of probability, a microstate refers to a specific arrangement of outcomes for a set of events. When discussing coin flips, each microstate represents a unique sequence of heads and tails resulting from the flips. When you flip three coins, each coin can land on either heads or tails, yielding a binary choice for each flip.
This means for each coin flip, there are two possible outcomes. To find the total number of microstates for multiple coin flips, you multiply the number of choices for each flip. Thus, the number of microstates from flipping three coins is calculated as: 

• There are 2 outcomes for the first coin
• There are 2 outcomes for the second coin
• There are 2 outcomes for the third coin

This multiplication gives us: \(2 \times 2 \times 2 = 8\). So, there are 8 possible microstates.

Coin Flipping and Probability

Coin flipping is a classic and simple example of a probabilistic event. Each flip of a fair coin has two possible outcomes: heads or tails, each with a probability of \(rac{1}{2}\). When we assign values to these outcomes, such as \(+1\) for heads and \(-1\) for tails, we start to explore how different sequences of flips (microstates) can result in various sums of these assigned values.
If you were to flip three coins, each face's outcome contributes to the overall sum of the microstate. For instance, if the sequence is HHT (Heads-Heads-Tails), the sum would be \(+1 +1 -1 = 1\). Understanding these outcomes helps in visualizing the distribution of probabilities across different sequences.

Sums of Outcomes in Microstates

The sum of outcomes assigns numerical values to a sequence of coin flips, reflecting the probability distribution of various sums stemming from these flips. The exercise illustrates how to calculate these sums based on the assigned values where heads is \(+1\) and tails is \(-1\). By listing all possible microstates for three coin flips, you can measure the frequency of each sum.
The outcomes for the sums discovered from microstates are:

• The sum of \(3\) occurs once, when all flips result in heads (HHH)
• The sum of \(1\) occurs in three different sequences (HHT, HTH, THH)
• The sum of \(-1\) also appears three times (HTT, THT, TTH)
• The sum of \(-3\) occurs once, when all flips are tails (TTT)

Thus, the most frequently occurring sums are \(+1\) and \(-1\), highlighting the central tendency of probabilistic outcomes in multiple coin flips.