There are 100 prizes, with one worth 1, one worth 2, …, and one worth 100. There are 100 boxe

Name: Problem 8, Chapter 3: Introduction to Probability
Uploaded: 2025-11-10T17:46:58-08:00
Duration: 1 min 9 s
Channel: Hoan Nguyen
Description: There are 100 prizes, with one worth 1, one worth 2, …, and one worth 100. There are 100 boxes, each of which contains one of the prizes. You get 5 prizes by picking random boxes one at a time, without replacement. Find the PMF of how much your most valuable prize is worth (as a simple expression in terms of binomial coefficients).

step 1 All right, solution. This i .J being 1 if price J is selected. 0 if not, then the total value. 1

Question

There are 100 prizes, with one worth $\$ 1$, one worth $\$ 2, \ldots$, and one worth $\$ 100$. There are 100 boxes, each of which contains one of the prizes. You get 5 prizes by picking random boxes one at a time, without replacement. Find the PMF of how much your most valuable prize is worth (as a simple expression in terms of binomial coefficients).

   There are 100 prizes, with one worth $\$ 1$, one worth $\$ 2, \ldots$, and one worth $\$ 100$. There are 100 boxes, each of which contains one of the prizes. You get 5 prizes by picking random boxes one at a time, without replacement. Find the PMF of how much your most valuable prize is worth (as a simple expression in terms of binomial coefficients).

Introduction to Probability

Joseph K.… 2nd Edition

Chapter 3, Problem 8

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Solved by Expert Hoan Nguyen

11/10/2025

Step 1

Then, X can take values from 1 to 100. We want to find the probability mass function P(X = k) for k = 1, 2, ..., 100. Show more…

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There are 100 prizes, with one worth $\$ 1$, one worth $\$ 2, \ldots$, and one worth $\$ 100$. There are 100 boxes, each of which contains one of the prizes. You get 5 prizes by picking random boxes one at a time, without replacement. Find the PMF of how much your most valuable prize is worth (as a simple expression in terms of binomial coefficients).

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Key Concepts

Order Statistics

This concept involves the study of the distribution of the maximum (or other order statistics) of a set of random variables sampled without replacement from an ordered collection. In the context of the problem, the maximum prize value is the largest number picked and its probability is derived by counting outcomes where one prize equals r and the other prizes are all less than r.

Combinatorial Analysis

The PMF is formulated through the use of binomial coefficients, a key tool in combinatorics that counts the number of ways to choose a subset from a larger set. In the given problem, these coefficients are used to count the ways to select the remaining prizes from the set of prizes less than r when determining the probability that the maximum prize is exactly r.

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