Statistics from Sciences: Probability

Question

During the Manhattan Project, the physicist Enrico Fermi asked Leslie R. Groves, the general in charge, "how do you define a 'great general'?" General Groves replied, "any great general who wins five battles in a row is great" He went on to say that only about 3% of generals are great. If battles are won entirely at random with a probability of 0.50 per side, what fraction of generals engaging in exactly five battles would be great by this definition? how does this compare to the percentage given by the general?

Answer 1

By chance, the number of generals winning 5 battles in a row (out of 5 battles, which is not the same as 5 in a row lifetime) is .5⁵ = 1/32 = 3.1 %. So , out of 100 generals, you'd expect 3.+ to win 5/5 battles. They are not necessarily great (just avg probability of winning). On top of that getting 5 in a row over a lifetime of battles is a much more likely probability (6 battles, 5 in a row is 3 x as likely)