Coin Tosses
Coin Tosses
What are the possible outcomes of tossing 5 coins? Each coin can land either heads or tails. On the large scale, we are concerned only with the total heads and tails and not with the order in which heads and tails appear. The following possibilities exist:
\(\begin{array}{c}\text{5 heads, 0 tails}\\ \text{4 heads, 1 tail}\\ \text{3 heads, 2 tails}\\ \text{2 heads, 3 tails}\\ \text{1 head, 4 tails}\\ \text{0 head, 5 tails}\end{array}\)
These are what we call macrostates. A macrostate is an overall property of a system. It does not specify the details of the system, such as the order in which heads and tails occur or which coins are heads or tails.
Using this nomenclature, a system of 5 coins has the 6 possible macrostates just listed. Some macrostates are more likely to occur than others. For instance, there is only one way to get 5 heads, but there are several ways to get 3 heads and 2 tails, making the latter macrostate more probable. The table below lists of all the ways in which 5 coins can be tossed, taking into account the order in which heads and tails occur. Each sequence is called a microstate—a detailed description of every element of a system.
5-Coin Toss
| Individual microstates | Number of microstates | |
|---|---|---|
| 5 heads, 0 tails | HHHHH | 1 |
| 4 heads, 1 tail | HHHHT, HHHTH, HHTHH, HTHHH, THHHH | 5 |
| 3 heads, 2 tails | HTHTH, THTHH, HTHHT, THHTH, THHHT HTHTH, THTHH, HTHHT, THHTH, THHHT | 10 |
| 2 heads, 3 tails | TTTHH, TTHHT, THHTT, HHTTT, TTHTH, THTHT, HTHTT, THTTH, HTTHT, HTTTH | 10 |
| 1 head, 4 tails | TTTTH, TTTHT, TTHTT, THTTT, HTTTT | 5 |
| 0 heads, 5 tails | TTTTT | 1 |
| Total: 32 |
The macrostate of 3 heads and 2 tails can be achieved in 10 ways and is thus 10 times more probable than the one having 5 heads. Not surprisingly, it is equally probable to have the reverse, 2 heads and 3 tails. Similarly, it is equally probable to get 5 tails as it is to get 5 heads. Note that all of these conclusions are based on the crucial assumption that each microstate is equally probable. With coin tosses, this requires that the coins not be asymmetric in a way that favors one side over the other, as with loaded dice. With any system, the assumption that all microstates are equally probable must be valid, or the analysis will be erroneous.
The two most orderly possibilities are 5 heads or 5 tails. (They are more structured than the others.) They are also the least likely, only 2 out of 32 possibilities. The most disorderly possibilities are 3 heads and 2 tails and its reverse. (They are the least structured.) The most disorderly possibilities are also the most likely, with 20 out of 32 possibilities for the 3 heads and 2 tails and its reverse. If we start with an orderly array like 5 heads and toss the coins, it is very likely that we will get a less orderly array as a result, since 30 out of the 32 possibilities are less orderly. So even if you start with an orderly state, there is a strong tendency to go from order to disorder, from low entropy to high entropy. The reverse can happen, but it is unlikely.
100-Coin Toss
| Macrostate | Number of microstates | |
|---|---|---|
| Heads | Tails | (W) |
| 100 | 0 | 1 |
| 99 | 1 | \(1\text{.}0×{\text{10}}^{2}\) |
| 95 | 5 | \(7\text{.}5×{\text{10}}^{7}\) |
| 90 | 10 | \(1\text{.}7×{\text{10}}^{\text{13}}\) |
| 75 | 25 | \(2\text{.}4×{\text{10}}^{\text{23}}\) |
| 60 | 40 | \(1\text{.}4×{\text{10}}^{\text{28}}\) |
| 55 | 45 | \(6\text{.}1×{\text{10}}^{\text{28}}\) |
| 51 | 49 | \(9\text{.}9×{\text{10}}^{\text{28}}\) |
| 50 | 50 | \(1\text{.}0×{\text{10}}^{\text{29}}\) |
| 49 | 51 | \(9\text{.}9×{\text{10}}^{\text{28}}\) |
| 45 | 55 | \(6\text{.}1×{\text{10}}^{\text{28}}\) |
| 40 | 60 | \(1\text{.}4×{\text{10}}^{\text{28}}\) |
| 25 | 75 | \(2\text{.}4×{\text{10}}^{\text{23}}\) |
| 10 | 90 | \(1\text{.}7×{\text{10}}^{\text{13}}\) |
| 5 | 95 | \(7\text{.}5×{\text{10}}^{7}\) |
| 1 | 99 | \(1\text{.}0×{\text{10}}^{2}\) |
| 0 | 100 | 1 |
| Total: \(1\text{.}\text{27}×{\text{10}}^{\text{30}}\) | ||
This result becomes dramatic for larger systems. Consider what happens if you have 100 coins instead of just 5. The most orderly arrangements (most structured) are 100 heads or 100 tails. The least orderly (least structured) is that of 50 heads and 50 tails. There is only 1 way (1 microstate) to get the most orderly arrangement of 100 heads. There are 100 ways (100 microstates) to get the next most orderly arrangement of 99 heads and 1 tail (also 100 to get its reverse). And there are \(1.0×{\text{10}}^{\text{29}}\) ways to get 50 heads and 50 tails, the least orderly arrangement. The table above is an abbreviated list of the various macrostates and the number of microstates for each macrostate.
The total number of microstates—the total number of different ways 100 coins can be tossed—is an impressively large \(1\text{.}\text{27}×{\text{10}}^{\text{30}}\). Now, if we start with an orderly macrostate like 100 heads and toss the coins, there is a virtual certainty that we will get a less orderly macrostate. If we keep tossing the coins, it is possible, but exceedingly unlikely, that we will ever get back to the most orderly macrostate. If you tossed the coins once each second, you could expect to get either 100 heads or 100 tails once in \(2×{\text{10}}^{\text{22}}\) years! This period is 1 trillion (\({\text{10}}^{\text{12}}\)) times longer than the age of the universe, and so the chances are essentially zero. In contrast, there is an 8% chance of getting 50 heads, a 73% chance of getting from 45 to 55 heads, and a 96% chance of getting from 40 to 60 heads. Disorder is highly likely.
This lesson is part of:
Thermodynamics