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markhboi
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can anyone help with this:
A particular type of electronic component for use in PCs is mass produced and subject to quality control checks since it is known that 5% of all components produced in this way are defective. The quality of a day's output is monitored as follows. A sample of 20 components is drawn from the day's output (which may be assumed to be large) and inspected for defective components. If this sample contains 0 or 1 defectives the day's output is accepted, otherwise it is rejected. If it contains more than 2 defectives the output is rejected. If the sample contains 2 defective a second sample of 20 is taken. If this sample contains 0 defectives the output is accepted, otherwise it is rejected.
Use the binomial distribution to calculate the probability of
(i) 0
(ii) 1
(iii) 2
defectives in a sample of 20.
Hence calculate the probability that the day's output is accepted.
Suppose that it is estimated that it costs £200 to inspect a sample of size 20. What is the expected cost of a day's sampling?
The binomial distribution is a probability distribution that describes the likelihood of a certain number of successes in a fixed number of independent trials, each with a binary outcome (e.g. success or failure).
To calculate the probability of 0, 1, or 2 defectives, you will need to know the number of trials, the probability of success on each trial, and the number of successes you are interested in. Then, you can use the binomial probability formula: P(x) = (nCx)(p^x)(q^(n-x)), where n is the number of trials, p is the probability of success, q is the probability of failure (1-p), and x is the number of successes you want to calculate the probability for.
Calculating the probability of defectives allows us to assess the quality of a product or process. By understanding the likelihood of having a certain number of defective items, we can determine if the process needs to be improved or if the product meets our quality standards.
The binomial distribution can be used to estimate costs by calculating the expected number of defectives and multiplying it by the cost of each defective item. This gives us an estimate of the total cost of defects that can be used for budgeting and decision-making.
No, the binomial distribution is specifically designed for binary outcomes (e.g. success or failure). If the outcomes are not binary, a different probability distribution, such as the Poisson distribution, may be more appropriate.