Question

The simplest way to shuffle cards is to take the top card and insert it at a random place in the deck, called top random insertion, and then repeat. We will consider a deck to be randomly shuffled once enough top random insertions have been made that the card originally at the bottom has reached the top and then been randomly inserted. If the deck has \(n\) cards, then the probability that the insertion will be below the card initially at the bottom (call this card \(B\) ) is \(1 / n\). Thus the expected number of top random insertions before \(B\) is no longer at the bottom is \(n\). Once one card is below \(B\), there are two places below \(B\) and the probability that a randomly inserted card will fall below \(B\) is \(2 / n .\) The expected number of top random insertions before this happens is \(n / 2 .\) The two cards below \(B\) are now in random order. Continuing this way, find a formula for the expected number of top random insertions needed to consider the deck to be randomly shuffled.

Short answer

Expect about \( n(\ln(n-1) + \gamma) \) random insertions.

Step-by-step solution

Analyze the Probability Structure

The problem describes a process where the goal is to move the initial bottom card, labeled as \( B \), to the top of the deck through a series of random insertions. Initially, the probability that a top card is inserted below \( B \) is \( \frac{1}{n} \) where \( n \) is the total number of cards.

Calculate the Expected Insertions for each Card

As each card is successfully inserted below \( B \), the probability changes. For the first card, it is \( \frac{1}{n} \) with an expected number of insertions being \( n \). For the second card, the probability changes to \( \frac{2}{n} \) and the expected number of insertions is \( \frac{n}{2} \). This pattern continues up to the second highest card, where the probability is \( \frac{n-1}{n} \) and the expected insertions are \( \frac{n}{n-1} \), creating \( n-1 \) insertions total.

Establish the General Formula

Recognize that as each subsequent card is inserted below \( B \), the necessary number of expected insertions follows: \[ n + \frac{n}{2} + \frac{n}{3} + \ldots + \frac{n}{n-1} \]. This resembles the harmonic series, excluding the last term \( \frac{n}{n} = 1 \), performing it for \( n-1 \) terms.

Derive and Simplify the Expected Value

The sum of the expected insertions forms a calculation of \( n \) times the harmonic number \( H_{n-1} \). The harmonic number \( H_{m} \) is approximated by the formula \( \ln(m) + \gamma \), where \( \gamma \) is the Euler-Mascheroni constant. Hence, the expected number of insertions simplifies to: \[ n \left( \ln(n-1) + \gamma \right) \].

Key concepts

Probability Theory

Probability theory helps us make sense of randomness. It's vital for problems like shuffling cards, where chance determines the outcomes. Here, probability is used to calculate how likely it is for a card to land in a specific position in the deck.
In our shuffling scenario, we focus on card \( B \), initially at the bottom. Each time we insert the top card into the deck, there's a probability that it will be placed below \( B \).
Initially, when all \( n \) cards are above \( B \), this probability equals \( \frac{1}{n} \). Once one card is below \( B \), the probability increases to \( \frac{2}{n} \), as there are now two spots below \( B \).
This gradual increase in the probability continues until \( B \) reaches the top position, emphasizing how probability guides us in predicting random events.

• Probability quantifies uncertainty.
• It changes as conditions shift during shuffling.

Expected Value Calculation

Expected value aids in predicting outcomes when dealing with uncertainties. It is a crucial concept in stochastic processes, which analyze systems subject to random changes. By applying it, we can estimate how many shuffles are needed for card \( B \) to no longer be at the bottom.
Initially, the expected number of shuffles before one card is placed below \( B \) is \( n \). This means, on average, it will take \( n \) top random insertions. As more cards are placed below \( B \), the expected number of insertions decreases. For the second card, it's \( \frac{n}{2} \), and this pattern continues.
The expected total number of insertions forms a series: \[ n + \frac{n}{2} + \frac{n}{3} + \ldots + \frac{n}{n-1} \]
This pattern is central to calculating outcomes in random processes and supports making informed predictions.

• Expected value acts like a weighted average over possible outcomes.
• It provides clarity in uncertainty.

Harmonic Series

A harmonic series is a sequence of numbers adding the reciprocals of integers. In our card shuffling exercise, it represents expected insertions.
Consider the sequence: \( 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n-1} \). These reciprocals form the harmonic series, which grows logarithmically.
When we multiply each term by \( n \), it reflects the expected number of insertions needed for each subsequent card position above \( B \). The total expected number of insertions then becomes \( n \) multiplied by the harmonic number, \( H_{n-1} \).
The harmonic number \( H_{m} \) can be estimated using \( \ln(m) + \gamma \), where \( \gamma \) is the Euler-Mascheroni constant.

• Harmonic series grow slowly.
• They are used in calculating processes involving probability.