What will be the probability space and the expected payment?

Question

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I found out the probability space to be (0.5)^x and the expected payment to be infinite.Still I have doubt in my answers.Please correct me If I am wrong.

typed question:

Stanley Statistics, an infamous statistician, wants you to enter a friendly wager with him. For $2,500, he will let you play the following game. He will continue to toss a fair coin until the first head appears. Letting x represent the number of the times coin was tossed to get the first head, Stanley will then pay you 2^x dollars.

a. Define a probability space for the experiment of observing how many times a coin must be tossed in order to observe the first head.

b. What is the expected payment that you will receive if you play the game?

c. Do you want to play the game? Why or why not?

Answer 1

The probability mass function describing the experiment is a Geometric distribution in the parametrization defined where $x \in \{1,2,3,...\}$

1. the probability space is a tern like $(Z,p(Z|\theta),\theta=\frac{1}{2})$ where $Z$ is the set of the possible results, $p(z)$ is the pmf that I showed you, and $\theta$ is the parameter of the fair coin to show H.

2.the expected number of tosses before the first H appears is obviously $x=2$ (read the link)

... I think you can conlcude by yourself if playing or not and why...