If customers arrive at a check-out counter at the average...

If customers arrive at a check-out counter at the average rate of $k$ per minute, then (see books on probability theory) the probability that exactly $n$ customers will arrive in a period of $x$ minutes is given by the formula $$P_{n}(x)=\frac{(k x)^{n} e^{-k x}}{n !}, n=0,1,2, \ldots$$ Find the probability that exactly 8 customers will arrive during a 30-minute period if the average arrival rate for this check-out counter is 1 customer every 4 minutes.

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Key Concepts

Poisson Distribution

The Poisson distribution is a discrete probability distribution that models the likelihood of a given number of events happening within a fixed interval of time or space, under the assumption that these events occur with a constant average rate and independently of one another. It is used extensively in scenarios involving count data or random arrivals.

Rate Parameter

The rate parameter represents the average number of occurrences per unit of time or space. In the context of the Poisson distribution, multiplying the rate by the length of the interval gives the expected count of events, which is central to calculating probabilities using the distribution.

Probability Mass Function

The probability mass function (PMF) for the Poisson distribution provides the probability of observing exactly n events in a fixed interval. It combines the rate parameter, the number of events, the exponential decay factor, and the factorial of n, encapsulating how likely it is to see a specific count of occurrences.

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