Probability and Combinatorics

Oct 13, 2025 by wbrum.org 132 views

Introduction

Clarence is a bread enthusiast with a bag full of loaves, and he's about to embark on a thrilling adventure of probability and combinatorics. With 7 small loaves and 3 large loaves, Clarence has a total of 10 loaves in his bag. He draws out a loaf uniformly at random, one-by-one, and we're tasked with finding the probability of a specific event. In this article, we'll delve into the world of probability, combinatorics, and expected value to solve Clarence's bread conundrum.

The Problem

Clarence has 7 small loaves and 3 large loaves of bread in his bag. He draws out a loaf uniformly at random, one-by-one. If he draws out a small loaf first, what is the probability that he will draw out a large loaf on his second draw?

Probability and Combinatorics

To solve this problem, we need to understand the concept of probability and combinatorics. Probability is a measure of the likelihood of an event occurring, and it's usually expressed as a value between 0 and 1. Combinatorics, on the other hand, is the study of counting and arranging objects in various ways.

In this case, we have 10 loaves in total, and we want to find the probability of drawing a large loaf on the second draw, given that we drew a small loaf first. To do this, we need to calculate the probability of drawing a small loaf on the first draw and then multiply it by the probability of drawing a large loaf on the second draw.

Calculating the Probability

Let's start by calculating the probability of drawing a small loaf on the first draw. There are 7 small loaves out of a total of 10 loaves, so the probability of drawing a small loaf on the first draw is:

P(Small on 1st draw) = 7/10 = 0.7

Now, let's assume that we drew a small loaf on the first draw. We're left with 8 loaves in the bag, and 3 of them are large loaves. The probability of drawing a large loaf on the second draw, given that we drew a small loaf first, is:

P(Large on 2nd draw | Small on 1st draw) = 3/8 = 0.375

Multiplying the Probabilities

To find the overall probability of drawing a large loaf on the second draw, given that we drew a small loaf first, we need to multiply the probabilities of the two events:

P(Large on 2nd draw | Small on 1st draw) = P(Small on 1st draw) × P(Large on 2nd draw | Small on 1st draw) = 0.7 × 0.375 = 0.2625

Expected Value

Expected value is a measure of the average value of a random variable. In this case, we can use expected value to find the average number of large loaves that Clarence will draw on his second draw, given that he drew a small loaf first.

Let's assume that Clarence draws a small loaf on the first draw. The expected value of the number of large loaves that he will draw on his second draw is:

E(X) = P(Large on 2nd draw | Small on 1st draw) × 1 = 0.2625 × 1 = 0.2625

Conclusion

In this article, we solved Clarence's bread conundrum by using probability and combinatorics to find the probability of drawing a large loaf on the second draw, given that we drew a small loaf first. We calculated the probability of drawing a small loaf on the first draw and then multiplied it by the probability of drawing a large loaf on the second draw. We also used expected value to find the average number of large loaves that Clarence will draw on his second draw, given that he drew a small loaf first.

Additional Questions

What is the probability of drawing a large loaf on the first draw?
What is the probability of drawing a small loaf on the second draw, given that we drew a large loaf first?
What is the expected value of the number of small loaves that Clarence will draw on his second draw, given that he drew a large loaf first?

Answers

P(Large on 1st draw) = 3/10 = 0.3
P(Small on 2nd draw | Large on 1st draw) = 7/9 = 0.7778
E(X) = P(Small on 2nd draw | Large on 1st draw) × 1 = 0.7778 × 1 = 0.7778

References

[1] Probability and Combinatorics by David M. Bressoud
[2] Expected Value by Investopedia

Note

Introduction

In our previous article, we solved Clarence's bread conundrum by using probability and combinatorics to find the probability of drawing a large loaf on the second draw, given that we drew a small loaf first. We also used expected value to find the average number of large loaves that Clarence will draw on his second draw, given that he drew a small loaf first. In this article, we'll answer some additional questions related to Clarence's bread conundrum.

Q&A

Q1: What is the probability of drawing a large loaf on the first draw?

A1: The probability of drawing a large loaf on the first draw is 3/10 = 0.3.

Q2: What is the probability of drawing a small loaf on the second draw, given that we drew a large loaf first?

A2: The probability of drawing a small loaf on the second draw, given that we drew a large loaf first, is 7/9 = 0.7778.

Q3: What is the expected value of the number of small loaves that Clarence will draw on his second draw, given that he drew a large loaf first?

A3: The expected value of the number of small loaves that Clarence will draw on his second draw, given that he drew a large loaf first, is E(X) = P(Small on 2nd draw | Large on 1st draw) × 1 = 0.7778 × 1 = 0.7778

Q4: What is the probability of drawing a small loaf on the first draw and then a large loaf on the second draw?

A4: The probability of drawing a small loaf on the first draw and then a large loaf on the second draw is P(Small on 1st draw) × P(Large on 2nd draw | Small on 1st draw) = 0.7 × 0.375 = 0.2625

Q5: What is the probability of drawing a large loaf on the second draw, given that we drew a small loaf first, and then a small loaf on the third draw?

A5: The probability of drawing a large loaf on the second draw, given that we drew a small loaf first, and then a small loaf on the third draw is P(Large on 2nd draw | Small on 1st draw) × P(Small on 3rd draw | Large on 2nd draw) = 0.2625 × 0.7 = 0.18375

Q6: What is the expected value of the number of large loaves that Clarence will draw on his second and third draws, given that he drew a small loaf first?

A6: The expected value of the number of large loaves that Clarence will draw on his second and third draws, given that he drew a small loaf first, is E(X) = P(Large on 2nd draw | Small on 1st draw) × 1 + P(Large on 3rd draw | Large on 2nd draw) × 1 = 0.2625 × 1 + 0.18375 × 1 = 0.44625

Conclusion

In this article, we answered some additional questions related to Clarence's bread conundrum. We calculated the probability of drawing a large loaf on the first draw, the probability of drawing a small loaf on the second draw, given that we drew a large loaf first, and the expected value of the number of small loaves that Clarence will draw on his second draw, given that he drew a large loaf first. We also calculated the probability of drawing a small loaf on the first draw and then a large loaf on the second draw, the probability of drawing a large loaf on the second draw, given that we drew a small loaf first, and then a small loaf on the third draw, and the expected value of the number of large loaves that Clarence will draw on his second and third draws, given that he drew a small loaf first.

References

[1] Probability and Combinatorics by David M. Bressoud
[2] Expected Value by Investopedia

Note

This article is for educational purposes only and is not intended to be used as a real-world solution to a problem. The calculations and examples used in this article are for illustrative purposes only and may not reflect real-world scenarios.