We’re playing a game. I flip a coin. If it lands on Tails, I flip it again. If it lands on Heads, the game ends.

You win if the game ends on an even turn, and lose otherwise.

Define the following events:

A: You win the game

B: The game goes on for at least 4 turns

C: The game goes on for at least 5 turns

What are P(A), P(B), and P(C)? Are A and B independent? How about A and C?

  • mathemachristian[he]@lemm.ee
    2·
    5 months ago
    solution

    Say Omega = N\{0}, sigma algebra is power set of N and the probability mass function is p(n)=2^-n .

    Then A is all the even numbers, B all numbers at least 4, C all numbers at least 5.

    P(A)=sum2-n n is even = sum2-2n n is at least 1 = sum4-n n is at least 1 = 1/(1-1/4)-1=1/3

    P(B) = P(N\{1,2,3}) = 1 - 1/2 - 1/4 - 1/8 = 1/8

    P© = 1/16 similarly

    P(A and B) = P(A\{2}) = 1/3 - 1/4 = 1/12 =/= P(A)P(B) therefore not independent

    P(A and C) = P(A\{2,4}) = P(A)P© with a similar calculation and therefore independent