### Educational Website Content---#### Probability and Combinatorics Exercises**Exercise 5: Probability in a Pizza Restaurant**In a pizza restaurant, 95% of the customers order pizza. If 65% of the customers order pizza and a salad, find the probability that a customer who orders pizza will also order a salad.**Exercise 6: Combinatorics Problem**How many different 3-digit identification tags can be made if the digits can be used more than once? If the first digit must be a 5 and repetitions are not permitted?**Exercise 7: Probability Distribution Verification**Verify the following table as a probability distribution. Draw the graph. Find the mean, variance, and standard deviation of the shown distribution.| x | P(x) ||------|------|| 2 | 0.1 || 4 | 0.4 || 6 | 0.3 || 8 | 0.2 |**Diagrams and Tables**- **Table Explanation:** The table represents a probability distribution with discrete random variable `x` and probabilities `P(x)` for each value of `x`. The values of `x` are 2, 4, 6, and 8, with corresponding probabilities 0.1, 0.4, 0.3, and 0.2, respectively.- **Graphing the Distribution:** To graph this probability distribution: 1. Plot `x` values on the horizontal axis (2, 4, 6, 8). 2. Plot `P(x)` values on the vertical axis. 3. Use bars to represent the probability of each `x` value.**Statistical Calculations:**- **Mean:** Calculate the expected value by summing $ x \times P(x) $ for each `x`.- **Variance:** Compute by summing $(x - \text{mean})^2 \times P(x)$ for each `x`.- **Standard Deviation:** Take the square root of the variance.---This educational content aims to enhance understanding of basic probability and combinatorics, with practical problems and statistical analysis.

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In a pizza restaurant, 95% of the customers order pizza. If 65% of the customers order pizza and a salad, find the probability that a customer who orders pizza will also order a salad.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Concept explainers

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.

Topic Video

Question

### Educational Website Content

---

#### Probability and Combinatorics Exercises

**Exercise 5: Probability in a Pizza Restaurant**

In a pizza restaurant, 95% of the customers order pizza. If 65% of the customers order pizza and a salad, find the probability that a customer who orders pizza will also order a salad.

**Exercise 6: Combinatorics Problem**

How many different 3-digit identification tags can be made if the digits can be used more than once? If the first digit must be a 5 and repetitions are not permitted?

**Exercise 7: Probability Distribution Verification**

Verify the following table as a probability distribution. Draw the graph. Find the mean, variance, and standard deviation of the shown distribution.

| x | P(x) |
|------|------|
| 2 | 0.1 |
| 4 | 0.4 |
| 6 | 0.3 |
| 8 | 0.2 |

**Diagrams and Tables**

- **Table Explanation:** The table represents a probability distribution with discrete random variable `x` and probabilities `P(x)` for each value of `x`. The values of `x` are 2, 4, 6, and 8, with corresponding probabilities 0.1, 0.4, 0.3, and 0.2, respectively.

- **Graphing the Distribution:** To graph this probability distribution:
1. Plot `x` values on the horizontal axis (2, 4, 6, 8).
2. Plot `P(x)` values on the vertical axis.
3. Use bars to represent the probability of each `x` value.

**Statistical Calculations:**

- **Mean:** Calculate the expected value by summing $ x \times P(x) $ for each `x`.
- **Variance:** Compute by summing $(x - \text{mean})^2 \times P(x)$ for each `x`.
- **Standard Deviation:** Take the square root of the variance.

---

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