## Question

###### Consider this gambling game. The player flips fair coin repeatedly until the player gets head, counting the number of flips required to get the head. If a head is achieved on the first flip, the player wins $2. If a head is achieved on the second flip (the Mlips went tails, heads), the player wins S4_ Ila head is achieved on the third flip (the flips went tails, tails, heads), the player wins $8_ The winnings continue the following this pallern. That is, if the first head is archived on the nth

Consider this gambling game. The player flips fair coin repeatedly until the player gets head, counting the number of flips required to get the head. If a head is achieved on the first flip, the player wins $2. If a head is achieved on the second flip (the Mlips went tails, heads), the player wins S4_ Ila head is achieved on the third flip (the flips went tails, tails, heads), the player wins $8_ The winnings continue the following this pallern. That is, if the first head is archived on the nth flip, then the player wins S2" What would you be willing t0 pay tO play this game? How many possible Outcomes are there t0 this game? Label the oulcomes 01,0z, 0= Whal are the values ol each outcome? We will now consider the probability of each outcome_ What is the probability that the firsi flip is a head? This IS the probability of the first oulcome Whal is the probability thal the player flips the coin (wice and gets tails first, then heads? This is the probability of the second oulcome Whal is the probability thal the player flips the coin three times and gets tails Gicsl; then tails again, then heads? This is the probability of the third outcome. Whal is the probability of each utcome? Whal the expecled value of this game? Now what would you be willing spend to play this game? What if the game is modified so that the first Outcome has value zero, now what is the expected value? What if the game is modified so that the firsl ten outcomes have value zero , now what is the expected value? Whal if the game is modified so that the first hundred oulcomes have value zero, now whal is the expecled value? What if the casino running this game has enough resources t0 make one thousand dollar payout, what is the actual expected value of the game? What if the casino running this game has enough resources to make one million dollar payout, what is the aclual expecled value of the game? What if the casino running this game has enough resources to make one billion dollar payout; what is the actual expected value of the gane?