Question

The hostess at a busy restaurant has found that during a Saturday evening, the restaurant typically has 14 diners entering every 18 minutes.

1. What is the probability that exactly 13 people enter the restaurant during a random 18 minute period on a Saturday evening?

2. What is the probability that exactly 10 people enter the restaurant during a random 9 minute window on a Saturday evening?

Answer 1

Part 1.

Let X denote the number of diners entering the busy restaurant every 18 minutes on a Saturday evening.

Now, since we are considering the number of events happening in a specific time interval we can use a Poisson distribution, thus we get:

X ~ Poisson(λ) for some λ > 0

=> E(X) = λ

Now, we are given that the restaurant typically (i.e., on an average) has 14 diners entering every 18 minutes during a Saturday evening, thus we get:

E(X) = 14

From the last two equations, we get:

λ = 14

=> X ~ Poisson(λ = 14) and the probability mass function of X is given by:

Thus, the probability that exactly 13 people enter the restaurant during a random 18 minute period on a Saturday evening is given by:

Part 2.

Let Y denote the number of diners entering the busy restaurant in a 9 minute window on a Saturday evening.

Now, since we are considering the number of events happening in a specific time interval we can use a Poisson distribution, thus we get:

Y ~ Poisson(µ) for some µ > 0

=> E(Y) = µ

Now, we are given that the restaurant typically (i.e., on an average) has 14 diners entering every 18 minutes during a Saturday evening, which means that in a 9-minute window on an average 7 diners enter on a Saturday evening. Thus we get:

E(Y) = 7

From the last two equations, we get:

µ = 7

=> Y ~ Poisson(µ = 7) and the probability mass function of Y is given by:

Thus the probability that exactly 10 people enter the restaurant during a random 9 minute window on a Saturday evening is given by:

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