Probability of All Orders Filled Correctly with 94.3% Accuracy

Math Problem Statement

Suppose that you and two friends go to a restaurant, which last month filled approximately

94.3 %94.3%

of the orders correctly. Complete parts (a) through (e) below.

Click the icon to view the order-filling data for two other restaurants.

Question content area bottom

Part 1

a. What is the probability that all three orders will be filled correctly?

The probability is

enter your response here.

(Round to four decimal places as needed.) Probability of Correct Order at Restaurant B

Data

Sample size

Probability of an event of interest

0.9090.909

Parameters

Mean

2.7272.727

Variance

0.24820.2482

Standard Deviation

0.49820.4982

Binomial Probabilities Table

P(X)

P(less than or equals≤X)

P(less than<X)

P(greater than>X)

P(greater than or equals≥X)

0.00080.0008

0.0000

0.99930.9993

1.00011.0001

0.02260.0226

0.02340.0234

0.00080.0008

0.97670.9767

0.99930.9993

0.22560.2256

0.24900.2490

0.02340.0234

0.75110.7511

0.97670.9767

0.75110.7511

1.00011.0001

0.24900.2490

0.0000

0.75110.7511

Probability of Correct Order at Restaurant C

Data

Sample size

Probability of an event of interest

0.8230.823

Parameters

Mean

2.4692.469

Variance

0.43700.4370

Standard Deviation

0.66110.6611

Binomial Probabilities Table

P(X)

P(less than or equals≤X)

P(less than<X)

P(greater than>X)

P(greater than or equals≥X)

0.00550.0055

0.0000

0.99450.9945

1.00001.0000

0.07740.0774

0.08290.0829

0.00550.0055

0.91710.9171

0.99450.9945

0.35970.3597

0.44260.4426

0.08290.0829

0.55740.5574

0.91710.9171

0.55740.5574

1.00001.0000

0.44260.4426

0.0000

0.55740.5574

Solution

a. What is the probability that all three orders will be filled correctly?

To find the probability that all three orders will be filled correctly at a restaurant that fills orders correctly 94.3% of the time, you can use the multiplication rule for independent events.

Given that each order has a 94.3% chance of being filled correctly, the probability that all three orders are filled correctly is:

$P(\text{all correct}) = P(\text{correct}) \times P(\text{correct}) \times P(\text{correct}) = (0.943)^3$

Let's calculate this:

$(0.943)^3 = 0.8391$

So, the probability that all three orders will be filled correctly is 0.8391.

Answer:

The probability is 0.8391 (rounded to four decimal places).

Do you want further details or have any questions?

Here are 5 related questions to deepen your understanding:

What is the probability that exactly two out of the three orders are filled correctly?
How would the probability change if the restaurant’s order-filling accuracy was 90% instead of 94.3%?
What is the probability that at least one of the three orders will be incorrect?
If five friends place orders, what is the probability that all five orders will be filled correctly?
How does the use of binomial distribution help calculate the probability in such problems?

Tip: In cases where the number of trials is small and the probability of success is known, the binomial distribution is a useful tool for calculating the likelihood of different outcomes.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Probability
Independent Events
Multiplication Rule for Probability

Formulas

P(all correct) = P(correct)^3 = (0.943)^3

Theorems

Multiplication Rule for Independent Events

Suitable Grade Level

Grades 9-12

Related Recommendation

Probability of Two Accurate Orders in Drive-Thru Selections with Replacement

Calculating Probability of Accurate Orders in a Drive-Thru with Replacement

Probability of Correct Orders at Multiple Restaurants: Analysis and Calculations

Probability of Accurate Orders in Fast Food Drive-Thru with Replacement

Calculate Probability of Not Accurate Orders from Fast Food Restaurants