Here's another attempt to show the "lumpiness" issue for different size investments. It's not easy to read exact values from this chart -- it's supposed to be illustrative:
On one horizontal axis you have the size of investment €10k, €20k, €30k ... €100k. On another you have the number of €50 wins in a year. On the vertical axis you have the percentage chance of getting that number of €50 wins for the given investment. For each size of investment, represented by a different colour, the highest bar is the most likely number of wins, and is close to the expected long run average number of wins. Of course, bigger investments result in bigger numbers of wins although the
percentage return eventually averages out the same for everyone.
It may seem paradoxical (or even
wrong!) that the highest percentage chance of hitting the average is for the smallest investment. But look closer. For that small investment, you also have a significant chance of only getting
half the average (22%), or nothing at all (10%)!
By contrast, look at the higher investments. In the following view I've swung the whole chart around so you can see the highest (€100k investment):
It's not easy to read the numbers, but the highest bar for the €100k investment represents 23 wins with a likelihood of about 8%. But you also have close to the same 8% chance each of getting 21 or 22 or 24 or 25 wins. That adds up to nearly a 40% chance of getting within 10% of your expected average return every year. By summing more bars we see an 80% chance that your return will be between 75% and 125% of the average and a better than 99% chance that it will be between 50% and 150% of the average. What are the odds of getting less than half the expected average? They are less than 0.3% ... vastly less likely than only getting half your expected average for a €10k investment.
Again, I stress that things will eventually average out. But it is likely to take
much longer to do so for small investments, possibly many years.