Question

At a Noodles & Company restaurant, the probability that a customer will order a nonalcoholic beverage is .56.

Answer 1

Answer:

a) P(X = 0) = 0.0000

b) P(X ≥ 4) = 0.9949

c) P(X < 5) = 0.0212

d) P(X = 15) = 0.0002

Step-by-step explanation:

We are given the probability that a customer will order a nonalcoholic beverage is 0.56

Note that there are only two possible outcomes i.e. either the customer will order a nonalcoholic beverage or he won't. The success in this case is the event that customer will order nonalcoholic beverage and probability of success stays the same i.e 0.56. The number of trials is fixed i.e. 15 and each trial is independent of the other. Therefore, this setting satisfied all the conditions of a Binomial Experiment. So, we will use Binomial Probability to answer the questions.

Part a) None out of 15 customer will order a nonalcoholic beverage:

The formula for calculating the Binomial Probability is:

$P(X = x) = ^{n}C_{x}(p)^{x}(q)^{n-x}$

Here, p = Probability of success = 0.56

q = Probability of failure = 1 - p = 1 - 0.56 = 0.44

x = Number of trails with success

Since, we want to calculate the none of the customer orders nonalcoholic beverage, so x will be 0 in this case.

Using the values in the formula, we get:

$P(X = 0)=^{15}C_{0}(0.56)^{0}(0.44)^{15}=0.00000445$

Thus, rounded to 4 decimal places, the the probability that in a sample of 15 customers, none of the 15 will order a nonalcoholic beverage is 0.0000

Part b) Atleast 4 will order

For this case we need to calculate the following probability:

P( X ≥ 4 )

Using the fact the sum of all probabilities of an event is equal to 1, we can write:

P( X ≥ 4 ) + P( X < 4 ) = 1

This means,

P( X ≥ 4 ) = 1 - P( X < 4 )

P( X < 4 ) means X can be 0, 1, 2, or 3

So,

P( X < 4 ) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

Calculating these values using the formula we used in previous part, we get:

P(X = 0) = 0.00000445

P(X = 1) = 0.000085

P(X = 2) = 0.000763

P(X = 3) = 0.00421

Thus,

P(X < 4) = 0.00000445 + 0.000085 + 0.000763 + 0.00421 = 0.0051

And

P( X ≥ 4 ) = 1 - P( X < 4 ) = 1 - 0.0051 = 0.9949

This means, he probability that in a sample of 15 customers, at least 4 will order a nonalcoholic beverage is 0.9949

Part c) Fewer than 5 will order

For this part we need to calculate the following probability:

P(X < 5)

P(X < 5) means sum of following probabilities:

P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X + 4)

We calculated all these probabilities in previous part except P(X = 4), which will be:

$P(X = 4)=^{15}C_{4}(0.56)^{4}(0.44)^{11}=0.0161$

Therefore,

P(X < 5) = 0.00000445 + 0.000085 + 0.000763 + 0.00421 + 0.0161 = 0.0212

So, the probability that in a sample of 15 customers, fewer than 5 will order a nonalcoholic beverage is 0.0212

Part d) All 15 will order

For this part our number of success will be 15. So x = 15 and we need to find:

P(X = 15)

Using the values in the formula of Binomial Distribution, we get:

$P(X = 15)=^{15}C_{15}(0.56)^{15}(0.44)^{0}=0.000167$

Thus, rounded to 4 decimal places, the probability that in a sample of 15 customers, all 15 will order a nonalcoholic beverage is 0.0002