A vending machine automatically pours soft drinks into cups. The amount of soft drink dispensed into a cup is normally distributed with a mean of 7.6oz and standard deviation of 0.4oz.

(a) Estimate the probability that the machine will overflow an 8-ounce cup. (Round your answer to two decimal places.)
(b) Estimate the probability that the machine will not overflow an 8-ounce cup. (Round your answer to two decimal places.)
(c) The machine has just been loaded with 848 cups. How many of these do you expect will overflow when served

Using the normal distribution, it is found that:

a) 0.1587 = 15.87% probability that the machine will overflow an 8-ounce cup.

b) 0.8413 = 84.13% probability that the machine will not overflow an 8-ounce cup.

c) 135 cups are expected to overflow when served.

In a normal distribution with mean and standard deviation , the z-score of a measure X is given by:

In this problem:

Item a:

The probability of an overflow is the probability of the machine having 8 oz or more, thus, it is 1 subtracted by the p-value of Z when X = 8.

has a p-value of 0.8413.

1 - 0.8413 = 0.1587.

0.1587 = 15.87% probability that the machine will overflow an 8-ounce cup.

Item b:

0.1587 probability of an overflow, so 1 - 0.1587 = 0.8413 = 84.13% probability that the machine will not overflow an 8-ounce cup.

Item c:

0.1587 out of 848, thus:

135 cups are expected to overflow when served.

A similar problem is given at link

Step-by-step explanation:

Since the amount of soft drink dispensed into a cup is normally distributed, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = amount in ounce of soft drink dispensed into cup.

µ = mean amount

σ = standard deviation

From the information given,

µ = 7.6oz

σ = 0.4 oz

a) The probability that the machine will overflow an 8-ounce cup is expressed as

P(x > 8) = 1 - P(x ≤ 8)

For x = 8,

z = (8 - 7.6)/0.4 = 1

Looking at the normal distribution table, the probability corresponding to the z score is 0.84

P(x ≤ 8) = 1 - 0.84 = 0.16

b) P(x< 8) = 0.84

c) when the machine has just been loaded with 848 cups, the number of cups expected to overflow when served is

0.16 × 848 = 136 cups