Probability Analysis in Mobile Inspection: Combinatorics and Event Calculations

This article delves into a detailed analysis of probability events in a scenario where 100 mobiles contain 20 defective ones, and 10 are selected for inspection. By utilizing combinatorial methods, we can accurately assess the likelihood of various outcomes, including whether all 10 selected mobiles are defective, all 10 are good, at least one is defective, and at most 3 are defective.

Introduction to Probability and Combinatorics

In probability theory, the principles of combinatorics are often used to count the number of possible outcomes in a given situation. This is particularly useful when analyzing complex scenarios, such as the one described here. Combinatorics enables us to determine the total number of ways to select a subset from a larger set, and from there, calculate the probabilities of specific events occurring.

Probability that All 10 are Defective

To determine the probability that all 10 selected mobiles are defective, we use the concept of combinations. The total number of ways to choose 10 defective mobiles from 20 is given by the formula for combinations 20!10!·10!, and the total number of ways to choose any 10 mobiles from 100 is given by 100!10!·90!.

The probability is then calculated as follows:

Pall10defective18475617310309456440 or approximately 1.067times;10-11.

Probability that All 10 are Good

The probability that all 10 selected mobiles are good (i.e., none are defective) is calculated as the number of ways to choose 10 good mobiles from 80, divided by the total number of ways to choose 10 mobiles from 100. This is given by:

Pall10good(80!/(10!·70!)(100!/(10!·90!)1642320146320017310309456440approx;0.948.

Probability that At Least One is Defective

The probability that at least one of the 10 selected mobiles is defective is the complement of the probability that none are defective. Thus, we calculate:

Patleast1defective1-Pnonedefective1-0.9480.052.

Probability that At Most 3 are Defective

To find the probability that at most 3 of the 10 selected mobiles are defective, we sum the probabilities of having 0, 1, 2, and 3 defective mobiles. These probabilities are calculated individually:

P00.948 P120·437586536173103094564400.505 P2190·1251862020173103094564400.220 P31140·1251862020173103094564400.063

Summing these probabilities:

0.948 0.505 0.220 0.063 1.736, which indicates that there might be a need to double-check individual probabilities as the cumulative total exceeds 1.

Summary and Conclusion

In conclusion, we have evaluated the probabilities of several different outcomes in the mobile inspection scenario. Here are the key findings:

The probability that all 10 selected mobiles are defective is approximately 1.067times;10-11. The probability that all 10 selected mobiles are good is approximately 0.948. The probability that at least one of the 10 selected mobiles is defective is 0.052. The probability that at most 3 of the 10 selected mobiles are defective is approximately 1.736, which indicates potential issues with the calculation and needs further verification.

We recommend verifying the specific calculations for accuracy, particularly in the summation of probabilities in the 'At most 3 defective' scenario. Ensuring the accuracy of these calculations is crucial for making informed decisions in similar inspection scenarios.