AMC 12B 2025 (Problem 18)
Awnik repeatedly plays a game that has a probability of winning of . The outcomes of the games are independent. What is the expected value of the number of games he will play until he has both won and lost at least once?
Let be the expected value of the number of games he will play until he has both won and lost at least once.
Let be the expected value of the number of games he will play until he has lost at least once.
Let be the expected value of the number of games he will play until he has won at least once.
Let be the expected value of the number of games he will play until he has lost at least once.
Let be the expected value of the number of games he will play until he has won at least once.
By Toolkit 44 — Creating n Equations in n Variables for Probability Problems, create 3 equations in terms of
From Hint 4,
From Hint 5,
Substitute these values into Hint 3.
From Hint 5,
Substitute these values into Hint 3.
(D)