11. If eight coins are thrown at random show that the probability of obtaining at least six heads is \( 37 / 256 \).
12. An apparatus contains six electronic tubes which all have to operate perfectly if it is to work. If the probability of each tube failing in a four hour period is 0.05 , what is the probability that the apparatus will fail in this intervad?
13. Three people, Alice, Bernard, and Charles, are playing a game for a prize. The first one to obtain heads when tossing a coin wins. They toss in succession starting with Alice, then Bernard, then Charles, et cetera. Show that Alice's chance of winning is \( 4 / 7 \). What are the chances of Bernard and of Charles winning?
14. A manufacturer knows that their resistors have values which are distributed as a Gaussian probability distribution with a mean resistance of \( 100 \Omega \) and standard deviation of \( 5 \Omega \).