During the Manhattan Project the physicist Enrico Fermi aske

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 coompare to the percentage given by the general?

Solution

Probability that a general is great means in any 5 random battles, he shouls win all of them in a row, i.e

lets take 5 battles where W represents win and L represents loss, according to general , a \'great generals\' scenario result of 5 battles should be like (W W W W W) , this only possible with a probability of 0.5*0.5*0.5*0.5*0.5 = (0.5)^5 = 0.03125 or 3.125%

But general estimated this to be 3% which is somewhat less than our calculated value 3.125%