Assignment:
Consider a state lottery that has a weekly television show. On this show, a contestant receives the opportunity to win $1 million. The contestant picks from 4 hidden windows. Behind each is one of the following: $150,000, $200,000 $1 million, or a "stopper". Before beginning, the contestant is offered $100,000 to stop. Mathematically speaking, should the contestant take the $100,000?
Suppose you toss a coin and will win $1 if it comes up heads. If it comes up tails, you toss again. This time you receive $2 if it comes up heads. If it comes up tails, toss again. This time you will receive $4 if it is heads. Continue in this fashion for a total of 10 flips of the coin, after which you receive nothing if it comes up tails. What is the mathematical expectation for this game?
Provide complete and step by step solution for the question and show calculations and use formulas.