Question

Eggs. A grocery supplier believes that in a dozen eggs, the mean number of broken ones is \(0.6\) with a standard deviation of \(0.5\) eggs. You buy 3 dozen eggs without checking them. a) How many broken eggs do you expect to get? b) What's the standard deviation? c) What assumptions did you have to make about the eggs in order to answer this question?

Short answer

Expected broken eggs: 1.8; Standard deviation: 0.866; Assumption: Independence.

Step-by-step solution

Understand the Problem

We need to calculate the expected number and standard deviation of broken eggs in 3 dozen, given the mean and standard deviation per dozen.

Calculate Expected Number of Broken Eggs

For one dozen eggs, the expected number of broken eggs is given as 0.6. Therefore, for 3 dozen eggs, the total expected number of broken eggs is calculated by multiplying the expected number per dozen by the number of dozens. \[Expected \: Broken \: Eggs = 0.6 \times 3 = 1.8\]

Calculate the Standard Deviation for 3 Dozen Eggs

The standard deviation for one dozen eggs is 0.5. Since standard deviations of independent events add up by the square root of the sum of squares, we calculate the combined standard deviation for 3 dozen eggs using the formula:\[Combined \: Standard \: Deviation = \sqrt{0.5^2 \times 3} = \sqrt{0.75} \approx 0.866\]

Assumptions

We assume that the number of broken eggs per dozen is independent and follows the same distribution with a mean of 0.6 and a standard deviation of 0.5. This implies each dozen of eggs fails independently with the same probability.

Key concepts

Expected Value

Expected value is a fundamental concept in probability that represents the average outcome of a random event if it were repeated many times. In the context of our exercise, we're dealing with broken eggs in several dozen. The mean number of broken eggs per dozen is given as 0.6. This is the expected value for a single dozen. To find the expected number for three dozen, we simply multiply the expected value per dozen by the number of dozens:

• One dozen eggs: 0.6 broken eggs expected
• Three dozen eggs: 0.6 (per dozen) × 3 (dozens) = 1.8 broken eggs expected

Therefore, when dealing with multiple groups or units, the expected value scales linearly. Always check if your random events, like broken eggs, are independent, as this assumption holds the math together.

Standard Deviation

Standard deviation measures the amount of variation or dispersion in a set of values. It tells you how much the number of broken eggs is likely to differ from the expected value. If the standard deviation is large, the results can vary widely from the expected value.For our problem, the standard deviation for one dozen eggs is 0.5. When considering the standard deviation for three dozen, since the dozen of eggs are independent from each other, we combine them using the square root of the sum of the squares:

• Standard deviation per dozen: 0.5
• Three dozen eggs: \[\sqrt{0.5^2 \times 3} = \sqrt{0.75} \approx 0.866\]

This approach ensures we account for all levels of variability, resulting in a better understanding of possible outcomes.

Independent Events

Independence in probability refers to scenarios where the outcome of one event does not affect another. In this exercise, assuming that the number of broken eggs in each dozen is independent is essential. Why? Because if the breaking of eggs was dependent, the calculations for expected value and standard deviation would need a different approach. Understanding independence helps with a few aspects:

• The calculations remain straightforward, relying on simple multiplication and use of standard formulas.
• It underpins the correctness of using the square root of the sum of squares for standard deviations.
• In real-world applications, it allows for more flexible planning and prediction.

When working with any statistical problem, ensure that your events are independent, unless otherwise indicated, to use these simplifying techniques.