Multiple Choice
Which of the following scenarios fits the condition of a process?
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4. Probability
Basic Concepts of Probability
Multiple Choice
If arrivals occur according to a distribution with an average of arrival every minutes, what is the probability that exactly arrivals occur in a -minute period?
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Verified step by step guidance1
Identify the rate parameter \( \lambda \) for the Poisson distribution over the given time period. Since the average arrival rate is 1 arrival every 20 minutes, for a 40-minute period, calculate \( \lambda = \text{rate} \times \text{time} = 1 \times 2 = 2 \).
Recall the Poisson probability mass function (PMF), which gives the probability of exactly \( k \) arrivals as:
\[
P(X = k) = \frac{\lambda^{k} e^{-\lambda}}{k!}
\]
Substitute \( k = 2 \) (since we want exactly 2 arrivals) and \( \lambda = 2 \) into the PMF formula:
\[
P(X = 2) = \frac{2^{2} e^{-2}}{2!}
\]
Calculate the factorial in the denominator: \( 2! = 2 \times 1 = 2 \). So the expression becomes:
\[
P(X = 2) = \frac{2^{2} e^{-2}}{2}
\]
This expression represents the probability of exactly 2 arrivals in 40 minutes. You can now evaluate this expression if needed.
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