Although the proportion of errors occurring in forensic DNA laboratories is low

Question

Although the proportion of errors occurring in forensic DNA laboratories is low due to regular proficiency testing, it is not zero. It thought that on average 30 laboratories commit errors in a re-accreditation time period. a. What is the probability 10 laboratories commit errors? b. What is the standard deviation in the number of laboratories that commit errors? c. What is the probability that more than 40 but less than or equal to 51 laboratories commit errors? d. If the probability of lab errors is 58%, up to how many laboratories committed errors? e. What is the probability that up to 35 but not less than 25 laboratories commit errors?

Explanation / Answer

Back-up Theory/concepts

This problem is strictly an application of Binomial Distribution. But, since the number of forensic DNA laboratories is very large, we can use Poisson Distribution.

If X is distributed as Poisson with parameter m [note that text books normally use the greek letter lembda instead of m], then P(X = x) = e-mmx/x!, where x any specific value that is a non-negative integer.

m is both the mean and variance of Poisson Distribution.

Now, to get the solutions, note that m = 30 in this case.

Part (a)

Required probability = P(X = 10) = e-303010/10!, This value can be obtained from Poisson Probability Tables.

Part (b)

As stated above, variance of Poisson Distribution is m. So, the standard deviation of the number of laboratories committing error = square root of 30 = 5.5 to be rouded off to 6 since number of laboratories cannot be a fraction.

Part (c)

Required probability = P(40 < X < 50) = Sum of  e-3030x/x!, x = 41, 42, 43, ......, 49 This value can be obtained from Poisson Probability Tables.

Part (d)

Suppose k laboratories commit errors, then the given statement, in probability language, is that probability X takes any inyeger value from 0 to k, both inclusive is 0.58.

Thus, we have: Sum of  e-3030x/x!, x = 0, 1, 2, 3, ......, k This value can be obtained by extrapolating k from the Table of Poisson Cumulative Probability.

Part (e)

Required probability = P(24 < X < 35) = Sum of  e-3030x/x!, x = 25, 26, 26, ......, 34 This value can be obtained from Poisson Probability Tables. Note that 'upto 35' cannot include 35 and 'not less than 25' implies 25 must be included.

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