Probability of Getting At Least Two Desired Books: A Combinatorial Mathematics Problem

Understanding the probability of selecting at least two desired books from a set of eight, where three specific books are chosen out of four randomly picked, involves applying combinatorial mathematics. This article delves into the intricacies of such a problem, breaking it down into manageable cases and providing a comprehensive solution.

Problem Statement

You are presented with a set of 8 books, among which 3 are your favorites. You are to randomly select 4 books. What is the probability that you pick at least 2 of your desired books?

Mathematical Formulation

Let's break down the problem using combinatorial mathematics. We will denote:

Total Books (b) 8 Desired Books (w) 3 Non-Desired Books (b-w) 5 Books Picked (r) 4

The total number of ways to choose 4 books from 8 can be calculated using the combination formula:

comb(n, k) n! / (k!(n-k)!)

Hence, the total number of ways to choose 4 books from 8 is:

Total ways comb(8, 4) 8! / (4!4!) 70

To find the probability of getting at least 2 desired books, we need to consider two cases: picking exactly 2 desired books and picking exactly 3 desired books.

Case Analysis

Case 1: Picking Exactly 2 Desired Books

Ways to choose 2 desired books from 3:

3! / (1!2!) 3

Ways to choose 2 non-desired books from 5:

5! / (3!2!) 10

Total ways for this case:

3 × 10 30

Case 2: Picking Exactly 3 Desired Books

Ways to choose 3 desired books from 3:

3! / (0!3!) 1

Ways to choose 1 non-desired book from 5:

5! / (4!1!) 5

Total ways for this case:

1 × 5 5

Adding the total ways from both cases:

30 (case 1) 5 (case 2) 35

Calculating the Probability

The probability of getting at least 2 desired books is given by the ratio of the total favorable outcomes to the total possible outcomes:

P(at least 2 desired) 35 / 70 0.5

This simplifies to 50%, indicating that the probability of picking at least 2 desired books when selecting 4 from a set of 8 is 50%.

Generalization

Problems like this can be generalized as follows:

Let B be a set of books, with a subset W of desired books and a randomly drawn subset R of B consisting of r books. Let:

b size of set B (total books) w size of set W (desired books) r size of set R (randomly drawn books) c number of common elements between W and R (wanted and picked books)

The number of choices for W and R is:

choose(b, w) choose(b, r)

The number of choices producing exactly c common elements of W and R is:

choose(b, c) choose(b-c, w-c) choose(b-w, r-c)

The probability formula is:

choose(b, c) choose(b-c, w-c) choose(b-w, r-c) / choose(b, w) choose(b, r)

Conclusion

By applying combinatorial mathematics, we can accurately determine the probability of picking at least two desired books from a set of eight, given three specific books are chosen out of four. This problem showcases the power of combinatorial mathematics in real-world scenarios, providing a framework for similar probability calculations.