Expected Winnings of Game Consisting of Base and Bonus Games

[Onestein]

Suppose we have a game that consists of a main game (guess a dice) and a bonus one (guess a coin toss). The rules that I think should be taken into consideration for the analysis are the following:​

• A player can play the bonus game only when they won the base game. So, if the player loss the main game, they are forbidden to play the bonus game.​
• The player may choose not to play the bonus game. So, they withdraw the main game winnings.​
• If the player choose to play the bonus game and they guess the coin toss, their winnings are doubled. Else, they lose everything and they are forbidden to continue the game.​
• If the player wins the first bonus game, they may choose to play the second bonus game or to withdraw their winnings. And so on.​
• No more than 4 bonus games are allowed to play.​

What is the expected winnings of a single game (the base game + the bonus game)? Consider only the winning base game.​

Is the following calculations are correct?

 

[stapel]

Suppose we have a game that consists of a main game (guess a dice) and a bonus one (guess a coin toss). The rules that I think should be taken into consideration for the analysis are the following:​

• A player can play the bonus game only when they won the base game. So, if the player loss the main game, they are forbidden to play the bonus game.​
• The player may choose not to play the bonus game. So, they withdraw the main game winnings.​
• If the player choose to play the bonus game and they guess the coin toss, their winnings are doubled. Else, they lose everything and they are forbidden to continue the game.​
• If the player wins the first bonus game, they may choose to play the second bonus game or to withdraw their winnings. And so on.​
• No more than 4 bonus games are allowed to play.​

What is the expected winnings of a single game (the base game + the bonus game)? Consider only the winning base game.​

Is the following calculations are correct?

View attachment 35943

What are your thoughts? What have you tried? How far have you gotten?

Please be complete, so the helpers can see where you're bogging down. Thank you!

Eliz.

 

[Onestein]

What are your thoughts? What have you tried? How far have you gotten?

Hello Eliz,

What I have tried is presented on the posted image (I'm using a dark theme add-on for my browser and didn't notice that the white background of the image makes it difficult to distinguish the image from the rest of the post. Sorry about that).

I thought I should calculate a number of all possible combinations and somehow account for the possibility that the player may refuse playing the bonus game. But I'm not quite sure how to properly to account for that.