Question

At a Noodles & Company restaurant, the probability that a customer will order a nonalcoholic beverage is .51.

Find the probability that in a sample of 10 customers, at least 7 will order a nonalcoholic beverage.

Answer 1

`P(x\geq 7)=0.1886`

Explanation:

The variable x, that said the number of customer that will order a nonalcoholic beverage in a sample of n customers follows a binomial distribution. Because we have n identical and independent events with a probability p of success and (1-p) of fail.

So, the probability that x customers will order a nonalcoholic beverage is:

`P(x)=(n!)/(x!(n-x)!)*p^(x)*(1-p)^(n-x)`

Where n is the size of the sample and p is the probability that a customer order a nonalcoholic beverage, so replacing the values, we get:

`P(x)=(10!)/(x!(10-x)!)*0.51^(x)*(1-0.51)^(10-x)`

Now, the probability that at least 7 will order a nonalcoholic beverage is equal to:

`P(x\geq 7)=P(7)+P(8)+P(9)+P(10)`

Where:

`P(7)=(10!)/(7!(10-7)!)*0.51^(7)*(1-0.51)^(10-7)=0.1267\\P(8)=(10!)/(8!(10-8)!)*0.51^(8)*(1-0.51)^(10-8)=0.0494\\P(9)=(10!)/(9!(10-9)!)*0.51^(9)*(1-0.51)^(10-9)=0.0114\\P(10)=(10!)/(10!(10-10)!)*0.51^(10)*(1-0.51)^(10-10)=0.0011`

So,
`P(x\geq 7)` is equal to:

`P(x\geq 7)=0.1267+0.0494+0.0114+0.0011\\P(x\geq 7)=0.1886`

Finally, the probability that in a sample of 10 customers, at least 7 will order a nonalcoholic beverage is equal to 0.1886