Question

Two people, X and Y, play a game alternately until one of them is successful or there are ten unsuccessful plays. Let Ei be the event of a success on the ith play of the game. Let WX, WY, and W0 be the events that X, Y, or neither wins, respectively. Write an expression for the occurrence of each of these events in terms of occurrences of the events Ei, 1 ≤ i ≤ 10.

Answer 1

Let the compliment event of E_i be denoted as E_i^c  $\forall$ 1$\leq\,$ i $\leq$10

CASE 1: X starts the game.   

P(WX) = P(E₁) + P(E₁^cE₂^cE₃) + P(E₁^cE₂^cE₃^cE₄^cE₅) + P(E₁^cE₂^cE₃^cE₄^cE₅^cE₆^cE₇) + P(E₁^cE₂^cE₃^cE₄^cE₅^cE₆^cE₇^cE₈^cE₉)

P(WY) = P(E₁^cE₂) + P(E₁^cE₂^cE₃^cE₄) + P(E₁^cE₂^cE₃^cE₄^cE₅^cE₆) + P(E₁^cE₂^cE₃^cE₄^cE₅^cE₆^cE₇^cE₈) + P(E₁^cE₂^cE₃^cE₄^cE₅^cE₆^cE₇^cE₈^cE₉^cE₁₀)

P(W0) = P(E₁^cE₂^cE₃^cE₄^cE₅^cE₆^cE₇^cE₈^cE₉^cE₁₀^c)

CASE 2: Y starts the game.   

P(WX) = P(E₁^cE₂) + P(E₁^cE₂^cE₃^cE₄) + P(E₁^cE₂^cE₃^cE₄^cE₅^cE₆) + P(E₁^cE₂^cE₃^cE₄^cE₅^cE₆^cE₇^cE₈) + P(E₁^cE₂^cE₃^cE₄^cE₅^cE₆^cE₇^cE₈^cE₉^cE₁₀)

P(WY) = P(E₁) + P(E₁^cE₂^cE₃) + P(E₁^cE₂^cE₃^cE₄^cE₅) + P(E₁^cE₂^cE₃^cE₄^cE₅^cE₆^cE₇) + P(E₁^cE₂^cE₃^cE₄^cE₅^cE₆^cE₇^cE₈^cE₉)

P(W0) = P(E₁^cE₂^cE₃^cE₄^cE₅^cE₆^cE₇^cE₈^cE₉^cE₁₀^c)