Suppose that you and two friends go to a​ restaurant, which last month filled approximately 90.4 % of the orders correctly. Complete parts​ (a) through​ (e) below.
Probability of Correct Order at Restaurant B
Data
Sample size
3
Probability of an event of interest
0.853
Parameters
Mean
2.559
Variance
0.3762
Standard Deviation
0.6134
Binomial Probabilities Table
X
​P(X)
​P(less than or equals​X)
​P(less than​X)
​P(greater than​X)
​P(greater than or equals​X)
0
0.0032
0.0032
0.0000
0.9969
1.0001
1
0.0553
0.0585
0.0032
0.9416
0.9969
2
0.3209
0.3794
0.0585
0.6207
0.9416
3
0.6207
1.0001
0.3794
0.0000
0.6207
Probability of Correct Order at Restaurant C
Data
Sample size
3
Probability of an event of interest
0.929
Parameters
Mean
2.787
Variance
0.1979
Standard Deviation
0.4449
Binomial Probabilities Table
X
​P(X)
​P(less than or equals​X)
​P(less than​X)
​P(greater than​X)
​P(greater than or equals​X)
0
0.0004
0.0004
0.0000
0.9996
1.0000
1
0.0140
0.0144
0.0004
0.9856
0.9996
2
0.1838
0.1982
0.0144
0.8018
0.9856
3
0.8018
1.0000
0.1982
0.0000
0.8018
What is the probability that all three orders will be filled​ correctly?


We can calculate the probability that all three orders will be filled correctly using the binomial probability formula:

\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

Where:
- \( n \) is the sample size (number of trials),
- \( k \) is the number of successes (in this case, all 3 orders are filled correctly),
- \( p \) is the probability of an event of interest (correct order filled),
- \( \binom{n}{k} \) is the binomial coefficient.

Here, we are asked for the probability that **all three** orders are filled correctly, so \( k = 3 \). We will calculate this probability for both Restaurant B and Restaurant C.

### Restaurant B:
- Sample size \( n = 3 \),
- Probability of a correct order \( p = 0.853 \).

Using the binomial formula for \( k = 3 \):

\[
P(X = 3) = \binom{3}{3} (0.853)^3 (1 - 0.853)^{3-3}
\]

Since \( \binom{3}{3} = 1 \) and \( (1 - 0.853)^{0} = 1 \), we only need to calculate:

\[
P(X = 3) = (0.853)^3 = 0.6207
\]

Thus, the probability that all three orders are filled correctly at Restaurant B is **0.6207**.

### Restaurant C:
- Sample size \( n = 3 \),
- Probability of a correct order \( p = 0.929 \).

Again, for \( k = 3 \):

\[
P(X = 3) = \binom{3}{3} (0.929)^3 (1 - 0.929)^{3-3} = (0.929)^3
\]

Calculating:

\[
P(X = 3) = (0.929)^3 = 0.8018
\]

So, the probability that all three orders are filled correctly at Restaurant C is **0.8018**.

### Summary:
- Probability for Restaurant B: **0.6207**
- Probability for Restaurant C: **0.8018**

Would you like more details on these calculations or to explore another part of the problem?

Here are 5 related questions:
1. What is the probability that none of the three orders are filled correctly?
2. What is the expected number of correct orders out of three for each restaurant?
3. What is the probability that exactly two of the three orders are filled correctly?
4. How does changing the probability of a correct order affect the overall probabilities?
5. How is the standard deviation related to the variability of correct orders?

**Tip:** The binomial distribution is useful when you're dealing with discrete events, such as the success/failure of orders. It's a great tool for predicting outcomes based on given probabilities.

Suppose that you and two friends go to a​ restaurant, which last month filled approximately 90.4 % of the orders correctly. Complete parts​ (a) through​ (e) below. Probability of Correct Order at Restaurant B and Restaurant C. What is the probability that all three orders will be filled​ correctly?

Explore the probability of receiving correct orders at Restaurants B and C using the binomial distribution. This problem involves calculating probabilities using a sample size of 3 with given success probabilities (p = 0.853 for Restaurant B and p = 0.929 for Restaurant C). Learn how to apply key probability formulas and statistical concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation