Consider an espresso stand with a single barista. Customers arrive to the stand at the rate of 28 per hour according to a Poisson distribution. Service times are exponentially distributed with a service rate of 35 customers per hour. The probability that there are exactly 3 customers in the system is Group of answer choices

The probability that there are exactly 3 customers in the system is 0.1024

Explanation:

Arrival rate (λ) = 28 customers/hour

Service rate (μ) = 35 customers/hour

Proportion of busy time of the server (ρ) = λ/μ

ρ = 28/35 = 0.8

Probability that there are exactly 3 customers in the system

(P₃) = ρ³(1 - ρ)

P₃ = 0.8³(1 - 0.8)

P₃ = 0.512 × 0.2

P₃ = 0.1024

Therefore, the probability that there are exactly 3 customers in the system is 0.1024

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