Compute expected value of simple gambling games

Q: Compute expected value of simple gambling games

This question evaluates a candidate's ability to compute expected value and reason about sequential probabilities in simple gambling scenarios, covering competencies in expectation calculation, net payoff/fair entry fee determination for a die roll, and first-success probability for alternating coin tosses.

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Question

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You are asked two probability/EV questions.

1) Dice game expected value / fair entry fee

A game works as follows:

You pay an entry fee of $c$ dollars to play.
You roll a fair six-sided die once.
If the die shows $k\in\{1,2,3,4,5,6\}$ , you receive $k$ dollars as your payout.

Questions

What is the expected payout (in dollars)?
What is the expected net profit (payout minus entry fee)?
What is the largest entry fee $c$ you should be willing to pay if you want the expected net profit to be $\ge 0$ ?

2) Alternating coin tosses: first head wins

Two players alternate flipping a fair coin. Player A flips first.

The first player to flip Heads wins immediately.

Question: What is the probability that Player A wins?

Compute expected value of simple gambling games

Quick Overview

1) Dice game expected value / fair entry fee

2) Alternating coin tosses: first head wins

Solution

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