Problem 1.23 You play a match against an opponent in which at each point either you or he/she serves. If you serve you win the point with probability p,, but if your opponent serves you win the point with probability p2. There are two possible conventions for serving: (i) alternate serves; (ii) the player serving continues until he/she loses a point. You serve first and the first player to reach n points wins the match. Show that your probability of winning the match does not depend on the serving convention adopted.

Robby wins 2 out of every 3 chess matches he plays with Marlene. What is the probability that he wins exactly 5 of the next 6 matches

Zane is ordering a sundae at a restaurant, and the server tells him that he can have up to five toppings: a cherry, hot fudge, caramel, peanuts, and butterscotch sauce. Since he cannot decide how many of the toppings he wants, he tells the server to surprise him. If the server randomly chooses which toppings to add, what is the probability that Zane gets just caramel and butterscotch sauce? Express your answer as a fraction or a decimal number rounded to four decimal places.

A study tracked a large group of people for 10 years. At the start of the study, 20% were heavy smokers (H), 30% were light smokers (L), and 50% were non-smokers (N). The interest of the study was whether the people in the study died during the study (D) or not die (Dc ). The probability that a light smoker died during the study was twice that of the probability of a non-smoker dying. Also, the probability that light smoker died during the study was half that of the probability of a heavy smoker dying. A randomly selected person from the study died during the study. What is the probability that this person was a heavy smoker (given they died, what is the probability they were a heavy smoker H)?