Cars arrive at a motorway filling station at a rate of 18 every quarter of an hour.
bability that at least 23 cars arrive in a quarter of an hour period.

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Cars arrive at a motorway filling station at a rate of 18 every quarter of an hour. bability that at least 23 cars arrive in a quarter of an hour period.

UpStudy AI · Accepted Answer

Let $$ X $$ be the number of cars arriving in a quarter of an hour. Since the arrivals follow a Poisson process with rate $$ \lambda = 18 $$, we have

$$
X \sim \text{Poisson}(18).
$$

The probability that at least 23 cars arrive is

$$
P(X \geq 23) = 1 - P(X \leq 22).
$$

Using the cumulative distribution function of the Poisson distribution, we can express

$$
P(X \leq 22) = \sum_{k=0}^{22} \frac{18^k}{k!} e^{-18}.
$$

Thus,

$$
P(X \geq 23) = 1 - \sum_{k=0}^{22} \frac{18^k}{k!} e^{-18}.
$$

An approximate evaluation of this probability (using either a calculator or statistical software) gives

$$
P(X \geq 23) \approx 0.144.
$$

So, the probability that at least 23 cars arrive in a quarter of an hour is approximately $$ 14.4\% $$.
The probability that at least 23 cars arrive in a quarter of an hour is approximately 14.4%.

Cars arrive at a motorway filling station at a rate of 18 every quarter of an hour.
bability that at least 23 cars arrive in a quarter of an hour period.

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