(3) Solve the following problems. (a) [2pts] There is a new reality show on television that highlights probability and gam- bling. In one of the episodes, there are five people, each sitting in a room by themselves, tossing a biased coin (P(head) = p). The first person to get a head will win five mil- lion dollars, and is sure to scream with joy (such being the nature of contestants on these shows). What is the distribution of the number of tosses until we hear a scream? Explain. Assume that the contestants toss the coins simultaneously, so (for example) 2 tosses means that each of them tossed twice; and we could have that two of the contestants scream at the same time. (Hint: Let X be the number of tosses until we hear a scream. What are the possible values that X could take? How would you compute the probability of the first of these values? Now write down the formula.) (b) [3pts] A random variable X follows Geometric(p). Take any two integers s and t such that s, t > 0. First, calculate P(X > s). Then, show that P(X 2 s + t|X > s) = P(X > t). We call this as a memoryless (forgetfulness) property of the geometric distribution