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Problem of the Week

Problem 5 (due Monday, April 8)

An investor in a casino is offered a choice of getting a return each time a certain game is played. The game is played by tossing $N$ times a fair coin and recording the sequence of heads (H) and tails (T). Let $h$ be the number of appearances of HH in the recorded sequence and let $t$ be the number of appearances of HT. For example, when $N=5$ and THHHT is recorded then $h=2$ and $t=1$. The investor can choose to either get $h$ cents each time the game is played, or to get $t$ cents each time the game is played. Which choice offers a better expected return?

Overview

Every other Monday (starting 01/22/24), we will post a problem to engage our mathematical community in the problem solving activity and to enjoy mathematics outside of the classroom. Students (both undergraduate and graduate) are particularly encouraged to participate as there is no better way to practice math than working on challenging problems. If you have a solution and want to be a part of it, e-mail your solution to Marcin Mazur (mazur@math.binghamton.edu) by the due date. We will post our solutions as well as novel solutions from the participants and record the names of those who've got the most number of solutions throughout each semester.

When you submit your solutions, please provide a detailed reasoning rather than just an answer. Also, please include some short info about yourself for our records.