Question

The probability that a customer will order a nonalcoholic beverage is .48b. Find the probability that in a sample of 12 customers, at least 5 will order a nonalcoholic beverage. (Round your answer to 4decimal places.)

Answer 1

Given:

probability of ordering non-alcoholic beverage = 0.48

probability of not ordering non-alcoholic beverage = 1 - 0.48 = 0.52

FInd: the probability that in a sample of 12 customers, at least 5 will order a nonalcoholic beverage.

Solution:

Recall the binomial probability formula.

$P(x)=nCr\times p^r\times q^{n-r}$

where

p = probability of success: 0.48

q = probability of failure: 0.52

n = the number of samples: 12

r = number of success (at least 5 which means not 1, 2, 3, or 4.

To determine the probability of having at least 5, let's calculate when r = 0, r = 1, r = 2, r = 3, and r = 4.

Let's start with r = 0 and solve.

$P(0)=_{12}C_0\times0.48^0\times0.52^{12}$$\begin{gathered} P(0)=1\times1\times0.000390877 \\ P(0)=0.000390877 \end{gathered}$

At r = 1,

$P(1)=_{12}C_1\times0.48^1\times0.52^{11}$$\begin{gathered} P(1)=12\times0.48^\times0.000751686 \\ P(1)=0.0043297 \end{gathered}$

Now, let's solve for r = 2.

$P(2)=_{12}C_2\times.48^2\times.52^{10}$$\begin{gathered} P(2)=66\times.2304\times.00144555 \\ P(2)=0.02198 \end{gathered}$

Moving on to r = 3.

$P(3)=_{12}C_3\times0.48^3\times0.52^9$$\begin{gathered} P(3)=220\times0.110592\times0.0027799 \\ P(3)=0.067636 \end{gathered}$

Then, lastly at r = 4.

$P(4)=_{12}C_4\times0.48^4\times0.52^8$$\begin{gathered} P(4)=495\times0.05308\times0.00534597 \\ P(4)=0.14047 \end{gathered}$

Let's now add the probability of getting r = 0, r = 1, r = 2, r =3, and r = 4 customers ordering a nonalcoholic beverage.

$P(0)+P(1)+P(2)+P(3)+P(4)$$0.0043297+0.02198+0.067636+0.14047=0.2344157$$0.000390877+0.0043297+0.02198+0.067636+0.14047=0.2348$

0.2348 is the probability of at most 4 customers ordering a non-alcoholic beverage.

Since the question is the probability of at least 5 customers ordering a non-alcoholic beverage which is the opposite of the at most 4 customers, then, let's subtract its probability from 1.

$1-0.2348=0.7652$

Therefore, the probability that in a sample of 12 customers, at least 5 will order a nonalcoholic beverage is approximately 0.7652.