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Compute expected value of simple gambling games

J.P. MorganData Scientist · Technical Screen
EasyData ScientistTechnical ScreenSoftware Engineering Fundamentals

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This problem tests your understanding of expected value, probability distributions, and the mechanics of independent trials in simple gambling games.

The Gambling Gauntlet

As a data scientist at J.P. Morgan, you are tasked with modeling two specific gambling scenarios to determine their risk profile. Please solve the following two independent sub-problems.

Part 1: The Weighted Die

You are playing a game with a single six-sided die. You pay an entry fee $F$ to roll the die once. If the result is a 6, you win $20. If the result is anything else, you win $0. What is the maximum entry fee $F$ you should be willing to pay if you want the game to have a non-negative expected profit? Provide the formula and the final numerical value.

Part 2: Alternating Coin Tosses

You and an opponent take turns flipping a fair, two-sided coin. You go first. The first person to flip a "Heads" wins the game.

  1. Calculate the probability that you win on your first turn.
  2. Calculate the probability that you win on your second turn (i.e., you get Tails, opponent gets Tails, then you get Heads).
  3. What is the total probability that you win the game?

Requirements

  • Provide the logic and the calculations for each part.
  • Use Python if necessary to define a function calculate_expected_value(p_win, win_amount) or calculate_total_win_probability(n_turns) to generalize your findings.

Example

If a game gives you a 1/2 chance to win $10 and a 1/2 chance to win $0, the expected value is (0.5 * 10) + (0.5 * 0) = 5. Your entry fee should be at most $5.

Hints

Hint 1

For Part 1, the expected value is the sum of (probability of outcome * value of outcome) for all possible outcomes.

Hint 2

For Part 2, express your probability as an infinite geometric series: P(win) = P(1st turn) + P(3rd turn) + P(5th turn) + ...

Hint 3

Recall that the sum of an infinite geometric series with first term 'a' and common ratio 'r' is 'a / (1 - r)' where |r| < 1.

Follow-up questions

  • How does the maximum entry fee in Part 1 change if the die is loaded such that a 6 appears with 25% probability?
  • If the game in Part 2 allows for an infinite number of turns, does the game ever end with a draw? Why or why not?
  • How would you simulate the coin toss game in Python to verify your theoretical probability using a Monte Carlo approach?