Answer Happy

A player of a video game is confronted with a series of 6 opponents. Assume that the results from opponents are independent, and that when the player is defeated by an opponent the game ends. Let X be the random variable defined by [0, if a player defeats all opponents X = i, if a player is defeated by (7-i)th-opponent. (a) Provide a table by putting the list all possible outcomes for the game of a player and the list of the range of the random variable side by side in a corresponding manner (write an outcome as a series consisting of the letters s and f, e.g. ssf or ss ssf, each to indicate that the player is defeated by 3rd and 5th opponents, respectively. You may think that s = success and ƒ = fail.). (b) If the probability of defeating of each one of the first 4 opponents is constant (say p), and the probability of a successful defeating of each one of the last 2 opponents is also constant (say q), find the formula for f₁, f₂, and f3 (in terms of p and q, and i) such that fi(p,q), if i=0 if i=1,2 P(X=i)=f₂(p,q,i), |f₂(p,q,i), if i=3,4,5,6. Show that the function f(i) = P(X = i), with i in the range of X, is a probability mass function. (c) Find p and q if only 80% of the players defeat the first 4 opponents, and only 50% defeat all opponents.