91Ó°ÊÓ

Assume that every time you buy a box of Wheaties, you receive one of the pictures of the \(n\) players on the New York Yankees. Over a period of time, you buy \(m \geq n\) boxes of Wheaties. (a) Use Theorem 3.8 to show that the probability that you get all \(n\) pictures is $$ \begin{aligned} 1 &-\left(\begin{array}{l} n \\ 1 \end{array}\right)\left(\frac{n-1}{n}\right)^{m}+\left(\begin{array}{l} n \\ 2 \end{array}\right)\left(\frac{n-2}{n}\right)^{m}-\cdots \\ &+(-1)^{n-1}\left(\begin{array}{c} n \\ n-1 \end{array}\right)\left(\frac{1}{n}\right)^{m} \end{aligned} $$ Hint: Let \(E_{k}\) be the event that you do not get the \(k\) th player's picture. (b) Write a computer program to compute this probability. Use this program to find, for given \(n,\) the smallest value of \(m\) which will give probability \(\geq .5\) of getting all \(n\) pictures. Consider \(n=50,100,\) and 150 and show that \(m=n \log n+n \log 2\) is a good estimate for the number of boxes needed. (For a derivation of this estimate, see Feller. \({ }^{26}\) )

Step by step solution

01

Understanding the Given Formula

The formula provided represents the probability of obtaining all \(n\) player's pictures after buying \(m\) boxes. It uses the principle of Inclusion-Exclusion by subtracting the probability of missing at least one of the player's pictures from 1.

02

Applying Inclusion-Exclusion Theorem

To find the probability of getting at least one of each picture, we define \(E_k\) as the event of missing the \(k\)-th picture. The Inclusion-Exclusion principle states that we sum the probabilities of each individual \(E_k\), subtract the joint probabilities of each pair of events \(E_k\), and so on, alternating signs. This is the pattern seen in the provided equation.

03

Deriving the Probability Expression

The probability expression involves alternating terms with binomial coefficients and fractions representing incomplete sets of pictures. For any subset of players missing, the probability is based on combinations of players and the likelihood of not obtaining a picture after \(m\) trials: \(\left(\frac{n-k}{n}\right)^m\).

04

Writing a Computer Program

To calculate this probability, write a program using a loop or recursion to evaluate the formula for all terms up to the \(n\)-th. The program should compute this probability for increasing \(m\) until it surpasses 0.5, signifying over 50% chance of having all pictures.

05

Estimating \(m\) Using Approximate Formula

Use \(m = n \log n + n \log 2\) to estimate the required number of boxes. Compare the results from the program and the estimation for \(n = 50, 100, 150\) to demonstrate its accuracy. This estimation derives from statistical mechanics and provides an efficient approximation.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory

Probability theory is a fundamental part of mathematics that deals with the concept of uncertainty and likelihood. It helps you understand how likely an event is to happen. In the context of the Wheaties boxes exercise, it is used to figure out the chance of getting all pictures of the New York Yankees players after buying a certain number of boxes. Here's how probability applies:

• The probability of a single event is defined as the number of successful outcomes divided by the total possible outcomes.
• In this scenario, you're interested in getting all pictures, which is considered a successful outcome.
• The formula used in the exercise derives from the principle of Inclusion-Exclusion, which calculates the probability of multiple events happening together by considering overlaps.
• As you collect more boxes, the probability of having all different pictures increases, especially when accounting for the random nature of which player's picture is in each box.

Probability theory offers tools to calculate such chances effectively, giving insights into how many boxes should be bought to have a high likelihood of completing the collection.

Combinatorial Mathematics

Combinatorial mathematics involves counting, arranging, and finding patterns within sets. It provides a way to systematically count all possible outcomes and combinations, which is crucial in solving the Wheaties boxes problem. Here's how it relates:

• In this exercise, combinatorial mathematics is used through the concept of binomial coefficients, denoted as \( \binom{n}{k} \).
• These coefficients help calculate how many ways you can select a certain number of items from a larger set, which in this case relates to the different ways pictures can go missing from your collection.
• The principle of Inclusion-Exclusion uses these coefficients to adjust for overcounting in the probability of not getting at least one of certain pictures.
• This combinatorial approach ensures a complete and accurate account of all possible ways pictures could be distributed across the boxes.

By using combinatorial mathematics, you can derive the formula that accurately predicts the likelihood of having a complete set of all pictures after purchasing several boxes.

Statistical Mechanics

Statistical mechanics is a branch of physics that deals with large collections of particles and their behaviors. It connects macro-level observations with micro-level behaviors. Although originally from physics, its principles can apply to a variety of fields, including solving probability problems like in this exercise:

• The estimation formula \(m = n \log n + n \log 2\) is derived using concepts from statistical mechanics.
• This approximation helps predict the number of boxes needed to likely receive all pictures, based on the characteristic behavior of random processes.
• Statistical mechanics assists in handling complex collections and deriving averages or norms from them, much like predicting when all pictures will be obtained.
• It manages probabilities in systems with many components, helping create approximations that align well with large numbers, akin to guessing the distribution of pictures in Wheaties boxes.

By borrowing concepts from statistical mechanics, the exercise offers a practical method to approximate the required number of boxes needed, showing the inter-disciplinary applications of these mathematical tools.