1) Let Pn be the probability that three consecutive heads will not appear in n tosses of a fair coin.
a) Show that Pn= 1/2 (Pn-1)+ 1/4 (Pn-2)+ 1/8 (Pn-3)
b) Compute the probability that there will be no three consecutive heads in the ?rst 8 tosses.
2) Suppose that tomorrow's weather-whether it will be wet or dry-will be the same as today's weather with probability p. Let us assume that the weather is dry today and let Pn be the probability that it will be dry n days later.
a) Show that P0 = 1 and Pn=(2p-1)P(n-1) + (1-p) for n>=1
b) Find Pn for n>=0.