A thought experiment in probability by Leonardo Barichello: two people are stranded on an island with only one banana to eat. To decide who gets it, they agree to play a game. Each of them will roll a fair 6-sided die. If the largest number rolled is a 1, 2, 3 or 4, then player 1 gets the banana. If the largest number rolled is a 5 or 6, then player 2 gets it. Which player has the better chance?

Assignment

In an extension to this thought experiment, both players roll a fair $$n$$-sided die ($$n \geq 4$$). The sides of the die are marked with 1 to $$n$$ eyes. The outcome of a roll with this die is represented as an integer (int) between 1 and $$n$$.

Player 1 wins the game if the roll of both players is less than or equal to an upper limit $$L \in \mathbb{N}$$ ($$1 \leq L \leq n$$) that has been fixed in advance. Otherwise, player 2 wins the game. Your task:

• Write a function has_won that takes three arguments: i) the outcome (int) of a roll of the die by player 1, ii) the outcome (int) of a roll of the die by player 2 and iii) the upper limit $$L$$ (int). The function must return a Boolean value (bool) that indicates whether player 1 has won the game.

• Write a function winning_outcomes that takes two arguments: i) the number of sides $$n$$ (int) of the die and ii) the upper limit $$L$$ (int). The function must return a tuple with two numbers (int): i) the number of possible outcomes from two rolls with an $$n$$-sided die where player 1 wins and ii) the number of possible outcomes from two rolls with an $$n$$-sided die where player 2 wins.

  Tip

  To determine the number of winning outcomes for both players, the function may go through all possible outcomes from two rolls with an $$n$$-sided die and determine who wins the game for each possible outcome.

• Write a function odds that takes two arguments: i) the number of sides $$n$$ (int) of the fair die and ii) the upper limit $$L$$ (int). The function must return a tuple with two numbers (float): i) the probability of player 1 winning the game and ii) the probability of player 2 winning the game. Both probabilities must be expressed as a percentage.

  Tip

  The probability (as a percentage) of a player winning the game is equal to the number of possible outcomes where the player wins the game, divided by the total number of possible outcomes, times 100.

• Write a function winner that takes two arguments: i) the number of sides $$n$$ (int) of the fair die and ii) the upper limit $$L$$ (int). The function must return an integer (int): the value 0 if both players have an equal chance of winning, the value 1 if player 1 has the better chance of winning, and the value 2 if player 2 has the better chance of winning.

Example

>>> has_won(3, 2, 4)
True
>>> has_won(1, 5, 4)
False

>>> winning_outcomes(6, 4)
(16, 20)

>>> odds(6, 4)
(44.44444444444444, 55.55555555555556)

>>> winner(6, 4)
2

Epilogue

What kind of probability are people talking about when they say something has "almost no chance" or it's "almost certain"? This chart¹ brilliantly explains visualizing the perceptions of probability.

Resources

• Sullivan P (2020). Is the Last Banana Game Fair? Mathematics Teacher 113(1), 33–38. ²