Question

A new study by Square Trade indicates that smartphones are \(50 \%\) more likely to malfunction than simple phones over a 3-year period. \({ }^{10}\) Of smartphone failures, \(30 \%\) are related to internal components not working, and overall, there is a \(31 \%\) chance of having your smartphone fail over 3 years. Suppose that smartphones are shipped in cartons of \(N=50\) phones. Before shipment \(n=10\) phones are selected from each carton and the carton is shipped if none of the selected phones are defective. If one or more are found to be defective, the whole carton is tested. a. What is the probability distribution of \(x\), the number of defective phones related to internal components not working in the sample of \(n=10\) phones? b. What is the probability that the carton will be shipped if two of the \(N=50\) smartphones in the carton have defective internal components? c. What is the probability that the carton will be shipped if it contains four defectives? Six defectives?

Short answer

Answer: To find the probability distribution of defective smartphones in a sample of 10 phones, calculate the probabilities using the hypergeometric distribution formula for each value of x (number of defective phones in the sample) from 0 to 10. For the probability of carton shipping, calculate the probabilities in the same manner but consider the given number of defective smartphones in the carton (2, 4, or 6 in this case) to find the probability that none of the sampled 10 phones are defective.

Step-by-step solution

a. Probability Distribution of \(x\)

First, let's calculate the probability \((p)\) of a phone having defective internal components: \(p = 0.31 \times 0.30 = 0.093\). Since there are \(50\) smartphones in a carton, the expected number of defective smartphones related to internal components not working is \(50 \times 0.093 = 4.65\). Now, let's find the probability distribution of \(x\). We need to use the hypergeometric distribution formula: \(P(x) = \frac{C(k,x) \cdot C(N-k, n-x)}{C(N, n)}\) where: \(N = 50\): total number of phones \(n = 10\): number of phones sampled \(k\) = number of defective phones (due to internal components not working) \(x\): number of defective phones in the sample Now we will compute the probability for different values of \(x\), that is \(x = 0\) to\(x=10\):

Calculating probabilities for each value of x

For each value of \(x\) from \(0\) to \(10\), we will calculate the probabilities using the above hypergeometric distribution formula and report the distribution.

b. Probability of Carton Shipping: 2 Defectives

For this part, we are given that in the whole carton, there are two defective smartphones due to internal components not working. We need to find the probability that none of the sampled \(10\) phones are defective, given there are 2 defective smartphones in the carton. Again, we will use the hypergeometric distribution formula: \(P(x=0) = \frac{C(2,0) \cdot C(50-2, 10-0)}{C(50, 10)}\). Calculate the probabilities and get the answer.

c. Probability of Carton Shipping: 4 and 6 Defectives

Now we need to find the probabilities that the carton will be shipped if it contains 4 defectives and 6 defectives. For 4 defectives: \(P(x=0) = \frac{C(4,0) \cdot C(50-4, 10-0)}{C(50, 10)}\) Calculate the probabilities and get the answer. For 6 defectives: \(P(x=0) = \frac{C(6,0) \cdot C(50-6, 10-0)}{C(50, 10)}\) Calculate the probabilities and get the answer. By calculating the above probabilities, we can determine the probability distribution of \(x\) and the probability that the carton will be shipped based on the number of defective smartphones.

Key concepts

Hypergeometric Distribution

The hypergeometric distribution is a statistical distribution used when the probability of success changes without replacement. It focuses on a finite population that consists of two types of objects. In this scenario, we have a fixed total number of smartphones in a carton, and we are interested in the number of defective phones due to internal components within a sample.The hypergeometric formula is given by:\[ P(x) = \frac{C(k,x) \cdot C(N-k, n-x)}{C(N, n)} \]where:

• \(N\) is the total population size, which represents the total number of phones in the carton (50 in this case).
• \(n\) is the sample size (10 phones are selected for testing).
• \(k\) is the total number of defective phones in the population.
• \(x\) is the number of defective phones actually found in the sample.

This distribution is crucial when determining the likelihood of finding a certain number of defective phones in a sample without replacement.

Defective Components

Defective components in the context of smartphones refer to internal parts that fail to function correctly. In the study, it is stated that internal malfunction is a contributor to smartphone failure. Specifically, there is a 30% chance that a smartphone failure is due to these internal component issues, coupled with the 31% overall failure rate of smartphones in three years.Calculating the expected number of defective smartphones due to internal problems involves multiplying these probabilities:\[ p = 0.31 \times 0.30 = 0.093 \]Hence, within a carton of 50 smartphones, the expected defective phones are:\[ 50 \times 0.093 = 4.65 \]This calculation defines our expectations regarding how many phones might potentially be unreliable in terms of internal components. Understanding these numbers helps us assess the risk of shipping potentially defective products.

Carton Sampling

Carton sampling is a process used to check the quality of a batch of products by testing a number of items from the batch. In this example, before a carton of 50 smartphones is shipped, 10 phones are sampled. The criterion for shipping is that none of these selected phones should be defective. The idea is to avoid shipping a carton if defects are found among the sampled phones. If any sampled phone is found to be defective, the entire carton undergoes further testing. This method is cost-effective because testing every smartphone would be time-consuming and expensive. By choosing an appropriate sample size and understanding probabilities through methods like the hypergeometric distribution, companies can make informed decisions about the reliability of their products before they reach the customer.

Smartphone Reliability

Smartphone reliability encompasses how likely a device is to function correctly over its expected lifespan. From the study, we know that smartphones have a higher chance of malfunctioning compared to simpler phones over a 3-year period. An essential factor in determining reliability is the probability of failure due to different issues, such as defective internal components. Companies need to maintain high standards for reliability to ensure customer satisfaction and to minimize returns or complaints. In this scenario, the probability distribution calculated helps predict the chances of finding defective phones based on sampling, which is essential for assessing the carton’s shipping eligibility. Ensuring overall reliability includes monitoring such statistics regularly and improving manufacturing quality to reduce failure rates.