Calculating the Probability of No Bad Eggs in an Omelette

Imagine you have a carton of eggs with a specific number of bad and good eggs. The task is to calculate the probability of obtaining a perfect omelette, consisting of three eggs, with no bad ones in it. This scenario involves a combination of basic probability concepts and combinatorics. Let's explore how to compute this probability step by step.

Step-by-Step Calculation

We start with a carton of eggs that contains a total of 12 eggs, which include 9 good eggs and 3 bad eggs. To make an omelette, 3 eggs are randomly selected from this carton. We want to find the probability of having no bad eggs in our selection.

Combinatorics Basics

Combinatorics is the branch of mathematics that deals with selecting, arranging, and counting objects. In this problem, we use combinations, which represent the number of ways to choose a subset of items from a larger set without regard to order.

The formula for combinations is given by:

[binom{n}{r} frac{n!}{r!(n-r)!}]

Where:

( n ) is the total number of items, ( r ) is the number of items to choose, ( ! ) denotes factorial, which is the product of all positive integers up to a given number (e.g., ( 5! 5 times 4 times 3 times 2 times 1 120 )).

Calculating the Total Number of Ways to Choose 3 Eggs

The first step is to find the total number of ways to choose 3 eggs out of the 12 in the carton. Using the combination formula:

[binom{12}{3} frac{12!}{3!(12-3)!} frac{12!}{3!9!} frac{12 times 11 times 10}{3 times 2 times 1} 220]

This gives us the total number of possible combinations when choosing 3 eggs from 12.

Calculating the Number of Ways to Choose 3 Good Eggs

Next, we want to find the number of ways to choose 3 good eggs from the 9 good eggs available. Using the combination formula again:

[binom{9}{3} frac{9!}{3!(9-3)!} frac{9!}{3!6!} frac{9 times 8 times 7}{3 times 2 times 1} 84]

Computing the Probability

To find the probability of choosing 3 good eggs (and thus no bad eggs), we divide the number of favorable outcomes by the total number of possible outcomes:

[text{Probability} frac{text{Number of ways to choose 3 good eggs}}{text{Total number of ways to choose 3 eggs}} frac{84}{220}]

To simplify this fraction:

[frac{84}{220} frac{21}{55}]

Therefore, the probability of selecting 3 good eggs, and thus making a perfect omelette, is 21/55 or approximately 0.3818 (or 38.18%).

Conclusion

This method showcases the application of combinatorics and basic probability theory to solve a practical problem. Understanding these concepts can help in many real-world scenarios, from baking an omelette to making informed decisions in various fields, such as finance and data analytics.

Related Keywords

probability omelette combinatorics