Question

At a noodles & company restaurant, the probability that a customer will order a nonalcoholic beverage is .33 find the probability that in a sample of 8 customers at least 5 will order a nonalcoholic beverage?​

Answer 1

0.0846 (3 s.f.) = 8.46% (3 s.f.)

Explanation:

We can model the given scenario as a binomial distribution.

Binomial distribution

`X \sim \text{B}(n,p)`

where:

• X is the random variable that represents the number of successes.
• n is the fixed number of independent trials.
• p is the probability of success in each trial.

Given the probability that a customer will order a non-alcoholic beverage is 0.33, and the number of customers is 8:

`\boxed{X \sim \text{B}(8, 0.33)}`

where the random variable X represents the number of customers who order a non-alcoholic beverage.

To find the probability that at least 5 customers will order a non-alcoholic beverage, we need to find P(X ≥ 5).

The complement rule of probability states that the probability of an event occurring is equal to one minus the probability of that event not occurring. Therefore:

`\text{P}(X \geq 5) = 1 - \text{P}(X \leq 4)`

We can use a calculator to calculate P(X ≤ 4) using the binomial cumulative distribution function (cdf). Note that the binomial cdf on a calculator will give you the sum of all the binomial probabilities for values of your random variable less than or equal to a given number (i.e. P(X ≤ x)).

Inputting the values of n = 8, p = 0.33 and x = 4 into the binomial cdf:

`\text{P}(X \leq 4) = 0.915427613...`

Therefore:

`\begin{aligned} \text{P}(X \geq 5) &= 1 - \text{P}(X \leq 4)\\&= 1 - 0.915427613...\\&= 0.0845723865...\\&=0.0846\;(3\;\sf s.f.)\end{aligned}`

So the probability that at least 5 customers from a sample of 8 customers will order a non-alcoholic beverage is 0.0846 (3 s.f.).