Probability of Drawing Black and Green Socks in Different Scenarios

Introduction

The problem at hand deals with the probability of drawing black and green socks from a single collection, and delves into various scenarios involving replacement or non-replacement of the socks. This article aims to clarify the different approaches and probabilities involved in this problem.

Problem Statement

The initial problem statement provided conflicting information about the number of green socks (4, 6, and 8). For simplicity, let's consider the scenario with 12 green socks, which is the highest number provided. Therefore, the collection will consist of:

4 black socks 6 blue socks 12 green socks

Title: How Likely Is It to Draw a Black and a Green Sock Twice?

Scenario 1: Same Sock for Both Draws (Replacement)

In this scenario, once a sock is drawn, it is placed back into the collection before the next draw.

Step 1: Probability of drawing a black sock first.

Number of black socks 4

Total number of socks 22 (4 black 6 blue 12 green)

Probability(Black) 4/22

Step 2: Probability of drawing a green sock on the second draw.

Number of green socks 12

Total number of socks 22 (since the black sock is replaced)

Probability(Green) 12/22

Combined Probability (Black and Green in this order) (4/22) * (12/22) 48/484 ≈ 0.10

Scenario 2: Green and Black in Different Orders

The probability of drawing a green sock first and then a black sock is calculated similarly:

Step 1: Probability of drawing a green sock first.

Probability(Green) 12/22

Step 2: Probability of drawing a black sock on the second draw.

Probability(Black) 4/22

Combined Probability (Green and Black in this order) (12/22) * (4/22) 48/484 ≈ 0.10

Total Probability (Either Order) 48/484 48/484 96/484 ≈ 0.20

Scenario 2: Different Socks for Both Draws and Non-Replacement

In this scenario, once a sock is drawn, it is not placed back into the collection before the next draw.

Step 1: Probability of drawing a black sock first.

4 black socks / 22 total 2/11

Step 2: Probability of drawing a green sock on the second draw.

12 green socks / 21 remaining (since one black sock is removed) 12/21 4/7

Combined Probability (Black and Green in this order) (2/11) * (4/7) 8/77 ≈ 0.10

Step 1: Probability of drawing a green sock first.

12 green socks / 22 total 6/11

Step 2: Probability of drawing a black sock on the second draw.

4 black socks / 21 remaining (since one green sock is removed) 4/21

Combined Probability (Green and Black in this order) (6/11) * (4/21) 24/231 ≈ 0.10

Total Probability (Either Order) 8/77 24/231 24/231 24/231 24/231 ≈ 0.10

Conclusion

To conclude, the probability of drawing a black and a green sock, either in this order or in the reverse, is approximately 0.20 in the scenario with replacement and about 0.10 in the scenario with non-replacement. The exact probability depends on the specific numbers of each type of sock in the collection and the rules of the drawing process.