Question

There is a well-known city (which will go nameless here) whose inhabitants have the reputation of enjoying a meal only if that meal is the best they have ever experienced in their life. Otherwise, they hate it. Assuming meal quality is distributed uniformly across a person's life, what is the expected number of times inhabitants of this city are happy with their meals?

Short answer

Expected number of happy meals is approximately \( \ln(n) + \gamma \).

Step-by-step solution

- Understand the Problem

The problem involves a city where inhabitants are only happy with a meal if it is the best they have ever had. We need to find the expected number of meals that make them happy over their lifetime, assuming meal quality is distributed uniformly.

- Define the Variables

Let the total number of meals in a person's life be denoted as \( n \). Each meal's quality is represented by a unique random value.

- Determine the Probability

For each meal to be the best meal so far, it must exceed the quality of all previous meals. Given that meal qualities are uniformly distributed, the probability that the \(k\)-th meal is the best so far is \( \frac{1}{k} \).

- Calculate the Expected Number

To find the expected number of best meals, sum the probabilities for all meals. The expected number is given by the sum of the series: \[ E[n] = \sum_{k=1}^{n} \frac{1}{k} \].

- Simplify the Summation

The sum \( \sum_{k=1}^{n} \frac{1}{k} \) is known as the Harmonic series, which approximately equals \( \ln(n) + \gamma \) as \( n \) grows large, where \( \gamma \) is the Euler-Mascheroni constant (approximately 0.577).

- Conclusion

For a large number of meals, the expected number of times an inhabitant is happy with their meal is approximately \( E[n] \approx \ln(n) + \gamma \).

Key concepts

Expected value

Expected value is a central concept in probability theory. It represents the average outcome of a random variable over many trials. Think of it as the 'long-term average' if you could repeat the process an infinite number of times. For instance, if you flip a fair coin, the expected value of getting heads is 0.5 because, over many flips, half will be heads.
In the problem, we're using expected value to determine how often people in the city are happy with their meals. Specifically, we're interested in how many meals rank as the best they've ever had.

Probability theory

Probability theory is the branch of mathematics concerned with the analysis of random phenomena. It enables us to model and predict the likelihood of different outcomes. Core concepts include random variables, distribution functions, and expected values.
In our exercise, we use probability theory to analyze meal satisfaction. First, we define the probability that a meal is the best so far. Then, we calculate the expected number of best meals over a lifetime using principles of probability theory.

Uniform distribution

A uniform distribution is a type of probability distribution where all outcomes are equally likely. Imagine rolling a fair die; each number from 1 to 6 has an equal chance of appearing, which means the distribution is uniform.
In our problem, meal quality is uniformly distributed. This means that any meal has an equal chance of being any quality, assuming quality is measured on a consistent scale. Because of this, the probability that each meal is the best so far follows a predictable pattern.

Harmonic series

The harmonic series is the sum of the reciprocals of the natural numbers. Mathematically, it is expressed as:
\[ \text{Harmonic Series} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \text{...} + \frac{1}{n} \text{ for n terms.} \]
In the problem, the harmonic series helps us find the expected number of happy meals for a person. Each term \( \frac{1}{k} \) in the sum represents the probability that the k-th meal is the best so far.

Euler-Mascheroni constant

The Euler-Mascheroni constant (denoted as \( \gamma \)) is a fundamental constant in mathematics that appears in analysis and number theory. Its approximate value is 0.577. It often comes into play when summing the harmonic series, particularly for large numbers.
In this exercise, the harmonic series approximation includes the Euler-Mascheroni constant:
\[ \text{Sum of Harmonic Series for large n} \approx \ln(n) + \gamma \]
Hence, the expected number of best meals, as the number of meals grows large, involves this constant.