Question

Suppose a computer chip manufacturer rejects $2\mathrm{\%}$ of the chips produced because they fail presale testing. a) What's the probability that the fifth chip you test is the first bad one you find? b) What's the probability you find a bad one within the first 10 you examine?

Short answer

(a) 0.0185; (b) 0.1829

Step-by-step solution

Understanding Geometric Probability

In part (a), we need to find the probability that the first defective chip appears at the fifth position. This describes a geometric distribution scenario, where the probability of finding the first defect after a certain number of successful attempts before it follows the formula: $$P(X=k)=(1-p{)}^{k-1}\cdot p$$where $p=0.02$ is the probability of a defective chip, and $k=5$ is the position where we find the first defective chip.

Calculate Probability for Fifth Chip

Plug the known values into the geometric probability formula for part (a): $$P(X=5)=(1-0.02{)}^{5-1}\cdot 0.02$$Now calculate:$$P(X=5)=(0.98{)}^4\cdot 0.02\approx 0.0185$$ Therefore, the probability is approximately 0.0185.

Understanding Cumulative Probability

In part (b), we want to find the probability of finding at least one defective chip within the first 10 examined. This means using the cumulative distribution function of a geometric distribution: $$P(X\le n)=1-(1-p{)}^n$$where $n=10$ and $p=0.02$.

Calculate Cumulative Probability for 10 Chips

Substitute the known values into the cumulative formula for part (b): $$P(X\le 10)=1-(1-0.02{)}^{10}$$Calculate:$$P(X\le 10)=1-(0.98{)}^{10}\approx 1-0.8171=0.1829$$ Thus, the probability is approximately 0.1829.

Key concepts

Probability

Probability is a fundamental concept in statistics that measures the chance of an event happening. When we talk about probability in the context of defective chips, we're measuring the likelihood of finding a defective chip among a batch of chips.
To compute this probability, we use the concept of a geometric distribution. This is particularly useful when we're looking for the first occurrence of an event, such as finding a defective chip. The formula for the probability of finding the first defective chip at the k-th trial is given by:

• $$P(X=k)=(1-p{)}^{k-1}\cdot p$$

Here, p represents the probability of finding a defective chip, and k denotes the number of tests conducted before finding the first defective chip.
Using this approach, if the probability

is low, it indicates that defective chips are rare, and vice versa.

Cumulative Distribution Function

The cumulative distribution function (CDF) in statistics is used to find the probability that a random variable is less than or equal to a certain value. In the context of defective chips, the CDF helps us in situations where we want to know the probability of finding at least one defective chip within a set number of trials.
The formula for the CDF of a geometric distribution is:

• $$P(X\le n)=1-(1-p{)}^n$$

Here, n is the number of tests, and p is the probability of a single defective chip event. This formula helps us understand how these probabilities accumulate over multiple trials.
When you find a defective chip in fewer tries, it increases the cumulative probability, indicating better chances of catching defects earlier.

Defective Chips

Defective chips are those that do not meet the quality standards set by manufacturers, resulting in them failing presale testing. They are a crucial factor in the electronics manufacturing process, as too many defective chips can lead to significant costs.
To manage quality, manufacturers need to understand the probability of defects. In our example, with a rejection rate of 2%, this means that 2 out of every 100 chips produced are expected to be defective. This statistic is vital for determining the quality control processes and testing frequencies required to ensure mostly defect-free products.
Understanding and calculating these probabilities help manufacturers keep defect rates under control and maintain product reliability, ultimately protecting their brand reputation and reducing wastage.