Boundary Binomial Probabilities

00:00 6 marks

Given a binomial random variable \(X\) representing the number of successes in \(n\) independent trials with success probability \(p\), answer the following:

Part a

4 minutes 3 marks

Question

Calculate \(P(X=k)\) for a binomial random variable when \(k = 0\) and when \(k = n\). Use the binomial probability formula \(P(X=k)= \binom{n}{k} p^k (1-p)^{n-k}\). Simplify your answers.

Hint Hint

Hint

Recall the definitions of binomial coefficients and powers when \(k=0\) and \(k=n\). Specifically, remember that \(\binom{n}{0} = 1\), \(p^0 = 1\), \(\binom{n}{n} = 1\), and \((1-p)^0 = 1\).

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Complete Answer

For \(k=0\), \(P(X=0)= (1-p)^n\); for \(k=n\), \(P(X=n)= p^n\).

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Part b

4 minutes 3 marks

Question

Interpret the meaning of the edge-case probabilities \(P(X=0)\) and \(P(X=n)\) in a quality control context where \(X\) is the number of defective items in a batch of \(n\) items, and \(p\) is the probability of an item being defective.

Hint Hint

Hint

Consider what it means for a batch to have zero defective items versus having all items defective in terms of production quality and process control.

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Complete Answer

In quality control, \(P(X=0) = (1-p)^n\) represents the probability that there are no defective items in a batch of \(n\) items. A high value of \(P(X=0)\) indicates a low defect rate and a well-functioning production process. Conversely, \(P(X=n) = p^n\) is the probability that all \(n\) items in the batch are defective. A high value of \(P(X=n)\) suggests a serious quality issue, indicating a process that is producing mostly defective items.

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