**Question: **Two players alternate flipping a coin that comes up heads with probability p. The first one to obtain a head is declared the winner. We are interested in the probability that the first player to flip is the winner. Before determining this probability, which we will call f (p), answer the following questions.

- Do you think that f (p) is a monotone function of p? If so, is it increasing or decreasing?
- What do you think is the value of limp→1 f (p)?
- What do you think is the value of limp→0 f (p)?
- Find f (p).

**Analytical Solution**

This problem is a differently put version of exercise 3.51. Section 4 has been answered in that exercise, please refer to that. I will answer the questions 1 through 3 from the image below. Since we are interested in the first player, being the winner, this also means that we are looking at the probability that the geometric is odd. Similarly, if we are interested in the probability that player 2 is the winner, we would compute the probability that the geometric variable is even.

Let us examine the top curve from the chart below to answer questions 1 through 3

- Answer: f(p) is monotonously increasing (as evident from the plot)
- Answer: The probability seems to be approaching the limit of 1.0 although f(p) is undefined when p = 1 (refer to 3.51) for details
- Answer: The probability seems to be approaching the limit of 0.5 although f(p) is undefined when p = 0 (again, refer to 3.51 for the actual formula for f(p))

**Simulation Solution**

Plots have been already done in exercise 3.51.

**Code**

N/A

End of the post 😉

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