91Ó°ÊÓ

Suppose that a computer chip company has just shipped 10,000 computer chips to a computer company. Unfortunately, 50 of the chips are defective. (a) Compute the probability that two randomly selected chips are defective using conditional probability. (b) There are 50 defective chips out of 10,000 shipped. The probability that the first chip randomly selected is defective is \(\frac{50}{10,000}=0.005 .\) Compute the probability that two randomly selected chips are defective under the assumption of independent events. Compare your results to part (a). Conclude that, when small samples are taken from large populations without replacement, the assumption of independence does not significantly affect the probability.

Step by step solution

01

- Total Probability for the First Chip

There are a total of 10,000 chips and 50 defective chips. The probability of selecting a defective chip on the first draw is \( P(D_1) = \frac{50}{10,000} = 0.005 \).

02

- Conditional Probability for the Second Chip Given First is Defective

If the first chip is defective, there are now 49 defective chips left out of 9,999 total chips. Therefore, the conditional probability that the second chip selected is also defective is \( P(D_2|D_1) = \frac{49}{9,999} \).

03

- Calculate the Combined Probability

The probability that both randomly selected chips are defective using conditional probability is \( P(D_1 \cap D_2) = P(D_1) \cdot P(D_2|D_1) = 0.005 \cdot \frac{49}{9,999} \).

04

- Compute the Exact Combined Probability

Calculate \[ 0.005 \cdot \frac{49}{9,999} = 0.005 \cdot 0.0049 \approx 0.0000245 \].

05

- Calculate Probability Assuming Independence

Under the assumption of independence, the probability of selecting two defective chips is \( P(D_1) \cdot P(D_2) = 0.005 \cdot 0.005 = 0.000025 \).

06

- Compare Results

Compare the results: The probability using conditional probability is approximately 0.0000245, while the probability assuming independence is 0.000025. Notice that the difference between them is very small.

07

- Conclusion on Independence Assumption

When taking small samples from large populations without replacement, the assumption of independence does not significantly affect the probability calculations, as the results are nearly identical.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory

Probability theory is the branch of mathematics concerned with probability, the analysis of random phenomena.

In this exercise, we are dealing with the probability of selecting defective chips from a shipment.
To solve problems like this, we can use principles from probability theory.

• The probability of an event is a measure of the likelihood that the event will occur.
• Probabilities range from 0 to 1, where 0 indicates impossibility and 1 indicates certainty.
• In our example, finding the probability of selecting defective chips involves counting methods and understanding of scenario-based calculations (like the first chip affecting the second chip or independent selections).

These principles are the foundation for solving real-world probability problems and predicting outcomes.

Independence Assumption

In probability, two events are independent if the occurrence of one does not affect the probability of the other.

This concept is crucial when calculating probabilities in different scenarios.

• If events are independent, the occurrence of one event does not change the probability of the other event occurring.
• For example, if you select a chip and then select another chip without looking at the first, theoretically, these two events are independent.
• In our exercise, we calculated the probability under the independence assumption, where \( P(D_1) = 0.005 \) and \( P(D_2) = 0.005 \), giving \( P(D_1 \cap D_2) = 0.005 * 0.005 = 0.000025 \).

However, in reality, if you do not replace the first chip, the events are not truly independent, as the first selection affects the remaining choices.

Sampling Without Replacement

Sampling without replacement means that once an item has been selected for our sample, it is not returned to the population before the next selection.

This affects the probabilities of subsequent selections.

• In our problem, selecting a defective chip first reduces the total number of chips and defective chips remaining.
• This changes the conditional probability of the second chip being defective.
• Initially, the chance of picking a defective chip is \( \frac{50}{10,000} = 0.005 \). If we pick a defective chip, now we have \( \frac{49}{9999} \) chance on the second pick.

Hence, we see from the calculations that considering dependence, the combined probability slightly decreases. However, with large populations and small samples, the difference is often negligible.