(a) [2pts] There is a new reality show on television that highlights probability and gambling. 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 million 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 s + t|X s) = P(X t). We call this as a memoryless (forgetfulness) property of the geometric distribution.