Question

The Lakeside Coffee Shop morning customer load follows a normal distribution. The mean number of customers is 60 and the standard deviation is 15.

- Determine the probability that the number of customers tomorrow will be 50.
- If the store gets more than 70 customers, they need to have a fourth staff member working. What is the probability that they will need that fourth person tomorrow?
- If the number of customers falls below 42, then two staff members are sufficient. What are the chances only two workers are required tomorrow?

Step 1

**The normal distribution:**

A continuous random variable *X* is said to follow normal distribution if the probability density function of *X* is,

Step 2

**a.**

**Find the probability that the number of customers tomorrow will be 50:**

Consider the variable lakeside coffee shop morning customer load as *X.*

It is given that the variable lakeside coffee shop morning customer load *X* is normally distributed.

Here, the mean number of customers is μ = 60 and standard deviation is σ = 15.

The given requirement is that the, number of customers tomorrow should be 50. That is, *X = *50.

*An important property of continuous distribution:*

The probability of a continuous random variable can be calculated for any given interval within the range of values of the random variable. The PDF, *f *(*X*) of a continuous random variable gives the “density” of the probabilities about the value *X*. It does not, at all, indicate the exact probability of the value *X* of the continuous random variable.

As the continuous random variable can take an infinite number of values within a given range, the probability of taking one particular value within the range is practically 0.

Hence, probability that the number of customers tomorrow will be 50 will be 0, that is, the probability that the continuous normal random variable *X* will take exactly the value 50 is *P*(*X* = 50) = 0.

Thus, the probability that the number of customers tomorrow will be 50 is **0.**

Step 3

**b.**

**Find the probability that the lakeside coffee shop need fourth staff person tomorrow:**

It is given that the lakeside coffee shop need fourth staff person if the number of customers to the shop exceeds 70.

The given requirement is that ...

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