In order to demonstrate the "Law of Large Numbers" we have run an experiment 4 times. The experiment is to take three dice, one red, one blue, and one green, and toss them a given number of times. These are "fair" dice; each has sides marked 1, 2, 3, 4, 5, and 6; the probability of getting any one value on any one roll of a die is 1/6. The first time we do the "experiment we roll the dice 44 times. The second version rolls the dice 216 times; the third rolls the dice 4400 times; the fourth rolls the dice 46,656 times.

Toss the three die 44 times. When the program did such an experiment it produced the results shown on the Toss 44 Times page. It may be useful to open that page in a new window so that you can scroll through it while you follow this commentary.

The report contains five tables:

1. Considering single die: For the number of tosses the program has recorded the number of times that each face of each die has "come up". In addition, the program computes the percent of times that each face on each die has "come up". Then the program looks at the three dice as if the color did not matter. In doing this the program has effectively three times the number of tosses. The table shows the total number of times that each face has come up on the total number of rolls, along with the associated percent. Finally, the table contains the theoretical percent of times that each face should come up.
2. Consider as raw pairs: This table shows, in the first column, the 36 possible results considering each possible pair of dice: the red and the blue, the red and the green, and the blue and the green. The program then shows, for the number of tosses, in this case 44 tosses, the number of times that each of the possibilities happened, along with the percent of times each happened. Thereafter, the program gives the count of the times that the specified value happened across all three pairs, and the corresponding percent. Finally, the program gives the theoretical percent that each value should show up.
3. Consider as sum of pairs: The third table looks at the sum of the values for each pair of dice. Thus, the possible sums range from 2 to 12. This table reports the number and percent of times that each sum happens for each pair of dice, along with the number and percent of times that each sum happens across all three possible pairings. Finally, the table gives the theoretical percent of times that each sum should happen.
4. Consider as raw triplet: For this table the program looks at and counts the number of times that each possible outcome happens. There are 216 such possible outcomes so the table has 216 rows of results. The program counts the number of times that each outcome happens and it reports that number and the percent of times that that the outcome happens. As in earlier tables, the final column in this table gives the theoretical percent for each possible outcome.
5. Consider as the sum of all three: For this final table, the program looks at the sum of all three dice. That sum could be as low as 3 and as high as 18. The program counts the number of times that each sum happens. It reports that count, the percent of times that the sum happens, and the theoretical percent of times that each sum should happen.

1. Considering single die: Even though we expect each face to happen about 1/6 of a time, there is a wide range in the distribution of values across the dice. For example, the Red die showed a 3 twelve times, the Blue die showed a 5 only 4 times and the Green die showed a 3 only 3 times. Now we know that we could not possibly get the same number of "showings" for each of the six faces because we had 44 tosses and even if after 42 tosses each face had come up 7 times we still have two more tosses and we clearly cannot spread them across all six values. The most even distribution would have four faces showing 7 times and the remaining two faces showing 8 times. 44 tosses is just too small to see the effect of the law of large numbers. However, it is worth noting that when we look in the "Total" column, which represents 132 tosses, the percent of times that each face shows up, though not exactly 1/6, is at least closer to the theoretic value.
2. Consider as raw pairs: Given that there are 36 possibilities for pairs of values, we know that with 44 tosses we will not have an even distribution of the results. The table not only shows that but it shows how our results are far from such an even distribution. For each pair of dice we see that there are many results that never happened. For example, the Red-Blue pair never came up as 1 1, 1 4, 2 4, 2 5, 3 5, 4 2, 5 5, 6 2, or 6 6. However, for that same Red-Blue pair 3 6 and 4 1 both happened 4 times. Again, as we should expect, in the Total column, where we are looking at the results of 132 tosses, we see that only the 2 5 pair did not happen at all. However, even in that Total column, we see a wide range of reported values, all the way up to 3 6 happening 9 times.
3. Consider as sum of pairs: In this table we will look at the differences between the theoretical percentage and the actual percentage. For each of the pairs of dice the 44 tosses produced results that wander pretty far from the theoretic value. Thus, the Red-Blue pair produced a sum of 5 almost 16% of the time whereas, in theory, such a sum should only come up just over 11 % of the time. Likewise, the Red-Green pair produced a sum of 10 nearly 16% of the time but theory suggests that such a sum should only happen 8.33% of the time. The Blue-Green pair produces a sum of 8 about 20.5% of the time, but in theory, we should only see a sum of 8 about 14% of the time. Again, looking at the Total columns, those results represent 132 tosses and given the results, though still somewhat different from the theoretic values, are closer to those values.
4. Consider as raw triplet: Once we get to this table, in our case of 44 tosses, we have 44 results being spread over 216 equally likely possible outcomes. Clearly some items will happen once and others will not happen at all. And, some items may happen more than once. In our example, 3 4 4 and 4 1 6 each happen twice, representing 4.5455% of the time whereas, in theory, such results should only happen 0.463% of the time.
5. Consider as the sum of all three: For this table we have our 44 results spread over just 16 possible results, ranging from 3 to 18. Again, because we only have 44 tosses, it is not strange to have some values that are far from the theoretic value. For example, a sum of 6 happened 4 times, representing about 9% of the tosses, whereas, in theory, we should get a sum of 6 only about 4.63% of the time.

Toss the three die 216 times. When the program did such an experiment it produced the results shown on the Toss 216 Times page. It may be useful to open that page in a new window so that you can scroll through it while you follow this commentary.

What do we see when we roll the three dice, 216 times?

1. Considering single die: Even with 216 rolls, we see quite a bit of variability in the number of times that each value "comes up" on the different die. Remember that each "face" has a probability of 1/6. Therefore, we "expect" that with 216 rolls we would have each face show up about 36 times. However, the Red die has 3 show up 42 times and 6 show up only 25 times. That gives a high percent of 19.444% and a low of 11.574% compared to the expected 16.667%. The Blue die ranges from 12.5% to 19.907%. The Green die ranges from 13.889% to 19.907%. When the program combines all of these, then we have effectively 648 rolls and now, in the Total column, we see that the range has narrowed to 14.969% to 17.747%.
2. Consider as raw pairs: In the raw pairs table we have 36 possible outcomes, each equally likely to occur. Thus, we expect to have 6 occurrences of each set of values. We see that we have The Red die with as few as 1 occurrence and as many as 12, the Bkue die with as few as 2 and as many as 11, and the Green die with as few as 2 and as many as 11. When the data is combined, again considering 614 rolls, while we would expect each of the values to show up 18 times, the most we see is 28 and the least is 8.
3. Consider as sum of pairs: Now the 216 rolls are spread across only 11 possible outcomes. The table shows that the experience with each pair of dice is different but that the percentage values for each pair of dice is not far off from the theoretic percentage.
4. Consider as raw triplet: Remember that there are 216 possible values in this table. Rolling our dice 216 times would make it possible to have exactly 1 instance of each of the possible outcomes. Clearly this did not happen. There are many possible values that never came up and there are other values, such as 2 3& 2 and 2 4& 4, that happen many times, specifically 4 times for each of those two. Again, even though 216 might be considered a large number, when we have 216 possible outcomes we need many more rolls to see the effect of the law of large numbers.
5. Consider as the sum of all three: Looking at the sum of the numbers on the top of the rolled dice, for all three dice, we can get the values 3 through 18. Thus, with 216 rolls, we will spread those results over the 16 possible answers. The table shows that we although we are close to the theoretic values, we still have some divergence from those values. In particular, the sum of the dice came up as 16 eleven times or 5.0926% of the time whereas we would have expected such a sum to only show up 2.7778% of the time. The important part of this table is to note that our percentages are closer to the theoretic values than we had when we only did 44 tosses.

Toss the three die 4400 times. When the program did such an experiment it produced the results shown on the Toss 4400 Times page. It may be useful to open that page in a new window so that you can scroll through it while you follow this commentary.

What do we see when we roll the three dice, 4400 times?

1. Considering single die: Rolling the Red die 4400 times we would expect to get about 733 occurrences of each of the 6 faces. The table shows that we did find approximately 733 occurrences of each face but there is still some variability. In fact, there will almost always be some variability. However, looking at the percentages, and comparing them to the theoretic value 16.667% we see that overall we are not far off. The Red die had results that went from 16.182% to 17.273%. The Blue die had a bit more variability, ranging from 14.818% to 17.477%. The Green die ranged from 15.34% TO 17.545%. Finally, ignoring the color so we are looking, in the Total column, at 13200 tosses, the percentages have closed in on the theoretic value. The observed Total percentages ranged from 15.92% to 17.167%.
2. Consider as raw pairs: For the raw pairs of dice we are spreading the 4400 rolls over 36 possible outcomes. Therefore, since each outcome is equally likely, we would expect about 122 occurrences of each outcome. We see in the table that there is a range of values but that range in pretty much centered about our 122. In fact, the percentages are starting to approximate the theoretic value of 2.778%.
3. Consider as sum of pairs: With the sum of pairs we have the 4400 spread over 11 possible outcomes. Looking at the theoretic column we see that these are not equally likely outcomes. However, since we have so many rolls, the percent of each outcome is not very far from the theoretic percent. This is even more true in the Total column where we are looking at 13200 rolls.
4. Consider as raw triplet: For the raw triplets, each value is equally likely. With 216 possible outcomes we would expect, for 4400 rolls, to see about 20 occurrences of each value. What we see is that 4400 is not large enough to blur the variability in our results. For example, 1 1& 5 only happens 9 times and 2 4& 2 happens 30 times, giving 0.2045% and 0.6818% respectively. The lower value is less than half the expected value of 0.463% while the upper value is close to half again as much as the theoretic value.
5. Consider as the sum of all three: On the other hand, looking at the sum of the three dice, 4400 tosses brings us pretty close to the theoretic percentages.

Toss the three die 46,656 times. When the program did such an experiment it produced the results shown on the Toss 46656 Times page. It may be useful to open that page in a new window so that you can scroll through it while you follow this commentary.

What do we see when we roll the three dice, 46,656 times?

1. Considering single die: Rolling the dice so many times, when we look at the frequency with which each face shows up we are not surprised to find that the percentages closely approximate the theoretic value 16.667%. This is true for the Red, the Blue, and the Green die. It is even more so when we look at the Total column which represents, in this case, 139,968 rolls.
2. Consider as raw pairs: Even spreading the values out over the 36 different raw pairs, we have such a large number of rolls that the results are quite close to the theoretic 2.778%. Again, the result shown in the Total column is even closer to that theoretic value.
3. Consider as sum of pairs: The eleven possible values for the sum of a pair of dice show percentages that are quite close to the corresponding theoretic values.
4. Consider as raw triplet: The values for the 216 possible raw triplets show a bit more variability than we saw in the earlier tables on this page. That is because we have spread the 46,656 values over the 216 possible results. Thus, the expected frequency of each result is 216 time. We observe, in the table, a frequency as low as 178 (for 5 1 2) and as high as 250 (for 4 6 6). This, along with the other tables on this page, really points out that when we talk about the law of large numbers we do not have just one value in mind for what constitutes a large number. We would need to run the experiment with a number of rolls well above 46,656 in order to get the observed percentages for the raw triplets to be closer to the 0.463% that is the theoretic probability for each value.
5. Consider as the sum of all three: Once we return to just looking at the sum of the three dice and we have the number of possible results shrink back to 16, the table shows that the 46,656 rolls is enough to have the observed percentages be quite close to the theoretic values.