Probability Analysis in Egg Selection: Recipies and Outcomes

The egg selection problem is a classic probability problem that can be applied in various scenarios, from cooking to vaccine efficacy testing. Recently, there was a question posed: out of 12 eggs in a refrigerator, 2 are bad, and 4 are chosen at random to make a cake. What is the probability that at least one of the chosen eggs is good? This article will explore the problem, its solution, and related probability concepts.

Problem Statement

Given 12 eggs, 2 are bad, and 4 eggs are chosen at random to make a cake. The question asks for the probability that at least one of the chosen eggs is good.

Solution Approach

To solve this problem, we can use the complement rule, which involves calculating the probability of the complementary event and then subtracting it from 1. In this case, the complementary event is choosing 4 eggs that are all bad.

Step-by-Step Explanation

Step 1: Total Eggs

Total eggs: 12 Bad eggs: 2 Good eggs: 10

Step 2: Choosing 4 Eggs

We need to find the probability that all 4 chosen eggs are bad. However, since there are only 2 bad eggs, it is impossible to choose 4 bad eggs. Therefore, the probability of choosing all 4 bad eggs is 0.

Step 3: Probability Calculation

Since the probability of all 4 eggs being bad is 0, we can use the complement rule to find the probability that at least one of the chosen eggs is good:

Probability of at least one good egg 1 - Probability of all 4 being bad 1 - 0 1

Conclusion

The probability that at least one of the chosen eggs is good is 1, or 100%.

Alternative Solution

An alternative solution involves explicitly calculating the number of favorable outcomes and unfavorable outcomes. Here is a step-by-step breakdown:

Total number of ways to choose 4 eggs from 12: 12C4 495

Number of ways to choose exactly 1 bad egg (2C1) and 3 good eggs (10C3): 2C1 * 10C3 2 * 120 240

Number of ways to choose exactly 2 bad eggs (2C2) and 2 good eggs (10C2): 2C2 * 10C2 1 * 45 45

Probability of at least one bad egg: (240 45) / 495 285 / 495 19 / 33 ≈ 0.575758

Probability of at least one good egg: 1 - 0.575758 0.424242

Discussion and Key Concepts

This problem demonstrates some key concepts in probability, including the complement rule, combinations, and the calculation of probabilities using total outcomes and favorable outcomes. It also highlights the importance of understanding the problem context (in this case, the limitation of choosing more bad eggs than are available).

Keywords: probability calculation, egg selection, good eggs