Mathematical Way to Choose a Toilet - Numberphile

Overview of the Problem

  • At music festivals, an abundance of toilets are often poorly maintained, presenting a dilemma of which toilet to choose.

  • The challenge is to determine when to stop trying toilets and select one based on hygiene and cleanliness.

  • Key restrictions are that the objective is to find the best toilet, and once a toilet is rejected, it cannot be chosen again.

Mathematical Framework

  • The problem can be mathematically modeled to optimize the selection process for toilets using a specific strategy.

  • Example illustrates a small festival scenario with 3 toilets labeled by their hygienic quality:

    • Toilet 1: Most hygienic (best choice)

    • Toilet 2: Moderately hygienic

    • Toilet 3: Least hygienic (worst choice)

Permutation of Options

  • The order of toilets varies in permutations; for 3 toilets, there are 6 possible arrangements (3 factorial).

  • The odds of choosing the best toilet (Toilet 1) first is quite low due to these permutations.

  • The strategy of rejecting the first toilet aims to improve the selection outcome but can lead to poor choices under certain conditions.

Case Studies with 3 Toilets

  • First Attempt: Reject Toilet 1, select Toilet 3 (worst outcome).

  • Second Attempt: Reject Toilet 1, then reject Toilet 3, must select Toilet 2.

  • Successful selections can yield the best toilet only 50% of the time (3 successes out of 6 attempts).

Increasing Complexity with More Toilets

  • When adding a fourth toilet, the permutations rise to 24 possibilities.

  • The successful selection ratio decreases significantly; only 11 out of 24 options lead to selecting the best toilet (~46%).

  • This downward trend in success illustrates the limitations of the initial strategy as the pool of options increases.

Determining the Optimal Strategy

  • For larger numbers of toilets, a revised approach is needed to increase the probability of success.

  • Analytical findings show that inspecting about 37% of toilets before making a selection maximizes the chances of finding the best toilet.

  • After evaluating this 37% threshold, the first toilet that surpasses the previous observed toilets should be chosen.

Practical Application Example

  • In a hypothetical situation with 100 total toilets, one should assess the first 37 toilets, then select the first one that exceeds all prior evaluations.

Broader Implications of the Problem

  • The toilet selection model is a practical illustration of the "Secretary Problem" in mathematics.

  • Similar methodologies apply to scenarios such as hiring the best candidate in a series of interviews or choosing an ideal romantic partner after dating several individuals.

Conclusion

  • This humorous approach to a relatable festival problem illustrates deeper mathematical insights and encourages exploration of decision-making strategies in various contexts, while reminding us of the importance of cleanliness in public facilities.