Problem 15 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

In a box containing 100 eggs, 10 eggs are rotten. The probability that out of a sample of 5 eggs none is rotten if the sampling is with replacement is (a) \(\left(\frac{1}{10}\right)^{5}\) (b) \(\left(\frac{1}{5}\right)^{5}\) (c) \(\left(\frac{9}{5}\right)^{5}\) (d) \(\left(\frac{9}{10}\right)^{5}\)

Step-by-Step Solution

Answer

The correct answer is (d) \(\left(\frac{9}{10}\right)^{5}\).

Step 1: Determine Total Probability of Selection

First, understand the total makeup of the box. There are 100 eggs, and 10 of these are rotten. That means, there are 90 good eggs. The probability of picking a good egg at random is the ratio of good eggs to total eggs.

Step 2: Calculate the Probability of Picking a Good Egg

Calculate the probability of picking one egg that isn't rotten: \[ P( ext{Good Egg}) = \frac{90}{100} = \frac{9}{10} \]

Step 3: Apply the Probability to Multiple Draws

Since the sampling is done with replacement, the probability of picking a good egg each time remains the same for all selections.

Step 4: Calculate the Total Probability for All 5 Eggs

For none of the 5 eggs to be rotten, all 5 must be good eggs. This is calculated as: \[ (\frac{9}{10})^5 \]

Step 5: Compare With Given Options

Comparing with the given options, option (d) matches our derived expression: \[ \left(\frac{9}{10}\right)^{5} \]

Key Concepts

Sampling with replacement

Sampling with replacement is a fundamental concept in probability and statistics. In simple terms, it means that when you select an item from a population, you put it back into the population before making the next selection. This is essential because it ensures that each selection is independent of the others. For instance, in our egg example, picking one egg and then returning it back ensures that each time, there are still 100 eggs from which to choose, keeping the probability consistent.
Using replacement:

• Ensures the sample size remains constant.
• Maintains the probability of each draw being the same.

This method contrasts with 'sampling without replacement,' where once an item is removed, it's not put back, causing the probabilities to change with each selection.

Independent events

In probability, an event is considered independent if the occurrence or non-occurrence of it does not affect the probability of another event. When sampling with replacement, as in the exercise, each time you select an egg, the events are independent. This is because the probability of picking a good egg stays constant in each draw.
For our scenario of picking eggs:

• Each selection of an egg does not alter the makeup of the remaining eggs.
• The preceding choices do not alter the outcome or likelihood of subsequent selections.

Independence is crucial for calculations, as it allows us to use multiplicative principles to find total probabilities across multiple events or draws.

Mathematical reasoning

Mathematical reasoning is the process of using logical thinking to solve problems. In probability exercises, like the one outlined, it's about breaking down a problem into comprehensible parts and systematically solving each step.
For example:

• First, understand the total population and how it is organized.
• Next, compute the probability of a single desired outcome (like picking a good egg).
• Then, see what method (with replacement) allows us to maintain that probability for consecutive selections.
• Finally, apply the logical principles, like multiplying probabilities for independent events, to find the overall chance of an event occurring over several trials.

This step-by-step approach ensures clarity and increases understanding, crucial for mastering probability and related mathematical topics.