Enrico Fermi and "Great" Generals. Introduction to probability.

Enrico Fermi, the famous physicist, met an army man who told him that some person is a great general. What makes a great general Fermi asked. Someone who wins 5 consecutive battles, the army man responded. What percentage of generals are great, Fermi asked. About 3-4% was the response. Fermi rejoined that if only 3-4% of generals are great, there are only lucky generals and no great generals. Let's see why. Suppose we have 100 generals of same skill. Let's say all battles are determined purely by luck, with a fair coin toss, and no skill whatsoever. We should expect an unlucky 50 of the 100 generals, shown in blue, will have lost the first battle. While the lucky remaining 50, in various colors, would have won the first battle. Prior to the second battle, we have lucky 50 generals who have won one battle. After the second battle, we should expect an unlucky 25 of them, shown in red, will have lost the battle. The remaining lucky 25, shown in various colors, would have won the second successive battle. Prior to the third battle, we have 25 lucky generals who have won two battles in a row. After the third battle, we should expect an unlucky 12 of them, shown in green, will have lost the battle. The remaining lucky 13, in yellow, gray and gold, would have won the third successive battle. Prior to the fourth battle, we have 13 lucky generals who have won three battles in a row. After the fourth battle, we should expect an unlucky 6 of them, shown in yellow, will have lost the battle. The remaining lucky 7, in gray and gold, would have won the fourth successive battle. Prior to the fifth battle, we have 7 lucky generals who have won four battles in a row. After the fifth battle, we should expect an unlucky 3 of them, shown in gray, will have lost the battle. The remaining lucky 4, in gold, would have won the fifth successive battle. We started with a 100 generals of same skill. Purely by luck 4 of them, or 4%, have won five battles in succession. So Fermi's reasoning is that if there's only 4% generals who have won five battles in a row, it could be explained just by luck and not skill. What we saw can be alternatively understood as follows. If the probability of winning a battle is 0.5, the probability of two consecutive wins is 0.5 times 0.5. The probability of three consecutive lucky wins is 0.5 times 0.5 times 0.5. Or, it is 0.5 to the power 3. The probability of n consecutive lucky wins is 0.5 to the power n. The probability of five consecutive lucky wins is 0.5 to the power 5 or 0.03125. And this was roughly the percentage of generals who were considered great. Key learnings. 1. Lots of activities in daily life, such as making and evaluating decisions, involve luck. To better understand the role of luck, we should have some idea of probability analysis. 2. In situations where there are only two possible outcomes, such as head or tail, win or lose, red or blue, it is useful to look at equal probability of one half and understand what would happen. 3. If probability of win is half, then probability of five consecutive wins is 0.5 raised to power 5, or equal to 3.125%. This roughly matched the of generals who were considered great. So perhaps it was just luck, as Fermi opined and followed up with has any of them won 10 consecutive battles? Source, Carl Sagan, The Demon Haunted World, Science as a Candle in the Dark. Picture from Wikipedia.