Problem 109 - Applied Statistics and Probability for Engineers

Question

A batch of 350 samples of rejuvenated mitochondria contains 8 that are mutated (or defective). Two are selected from the batch, at random, without replacement. (a) What is the probability that the second one selected is defective given that the first one was defective? (b) What is the probability that both are defective? (c) What is the probability that both are acceptable?

Step-by-Step Solution

Answer

(a) \( \frac{7}{349} \). (b) \( \frac{56}{121975} \). (c) \( \frac{116532}{122150} \).

Step 1: Understanding Conditional Probability for Part (a)

For part (a), we need the probability that the second sample is defective given that the first is defective. Knowing that the first chosen is defective means there are 7 defective mitochondria left out of a total of 349 remaining samples. Use the formula for conditional probability, which is \( P(B|A) = \frac{P(A \cap B)}{P(A)} \). However, here it simplifies to simply counting from reduced totals: \( P(\text{second defective} | \text{first defective}) = \frac{7}{349} \).

Step 2: Calculating Probability for Part (b)

For part (b), both samples being defective corresponds to drawing a defective sample first and then another defective. Use the multiplication rule of independent probabilities modifying for without replacement: \( P(\text{both defective}) = P(\text{first defective}) \times P(\text{second defective | first defective}) = \frac{8}{350} \times \frac{7}{349} \). Simplify this expression to get the probability value.

Step 3: Calculating Probability for Part (c)

For part (c), calculate the probability of both samples being non-defective (acceptable). First, find the probability that the first is acceptable: \( P(\text{first acceptable}) = \frac{342}{350} \) since there are 342 non-defective samples. Then, if the first is acceptable, there are 341 acceptable samples left, so \( P(\text{second acceptable | first acceptable}) = \frac{341}{349} \). Multiply these probabilities to get \( P(\text{both acceptable}) = \frac{342}{350} \times \frac{341}{349} \).

Key Concepts

Independent Probabilities

When discussing probability, we often encounter the concept of independence. Two events are said to be independent if the occurrence of one does not affect the probability of the other occurring. In simpler terms, if knowing the outcome of one event gives you no information about the outcome of another, they are independent.

In the context of the exercise, however, this independence does not apply since the samples are selected without replacement. When we choose without replacement, the probabilities change based on previous selections.

• For example, if you pick a defective sample first, there are fewer defective samples left for the second pick.
• Thus, while normally two draws from a large population could be independent, drawing from a small batch without replacement makes the probabilities dependent on each selection.

This concept forms an important foundation in understanding the conditional probabilities used in the exercise.

Multiplication Rule

The multiplication rule is a fundamental principle used to find the probability of two events occurring in sequence. Generally, if two events are independent, the probability of both occurring is the product of their individual probabilities. However, as seen in our exercise, drawing from a limited batch without replacement alters this approach.

For instance, in part (b) of the exercise, we want the probability that both samples selected are defective. This required applying the multiplication rule considering the reduced samples—

• First, calculate the probability of drawing a defective sample initially.
• Then, calculate the conditional probability of selecting a defective sample again given the first was defective.

Thus, the correct form of the multiplication rule, when events are dependent like this, is to multiply the probability of the first event by the conditional probability of the second event given the first has occurred:
\[ P(A \text{ and } B) = P(A) \times P(B|A) \]
This adjusted multiplication rule is essential for accurate probability calculations when sampling without replacement.

Probability of Acceptable Samples

In probability exercises involving multiple possible events, like measuring acceptable samples, identifying what you need helps tremendously. In this exercise, we needed to find the probability of selecting two acceptable mitochondria, where acceptable means they are not defective.

First, you determine the probability of choosing one acceptable sample. Since 342 out of 350 are acceptable, the probability for the first draw is quite high.

• As soon as one sample is picked, the total number of remaining samples decreases, affecting subsequent probabilities.
• Thus, for the second sample, only 341 acceptable ones are left out of 349.

To find the probability that both samples are acceptable, multiply the probability of the first acceptable sample by the probability of getting a second acceptable one given the first: \[ P(\text{both acceptable}) = P(\text{first acceptable}) \times P(\text{second acceptable | first acceptable}) \]
This approach reiterates how changes in sample size affect probability calculations, particularly in the context of conditional probabilities when sampling without replacement is involved.