https://file.aiquickdraw.com/mathbot/file/1726723001368i2oh9cw4.jpg

The image contains a problem about calculating the length and the success probability of the following Bernoulli processes:

1. **(a)** A coin is flipped five times. The number of times heads comes up is counted.
2. **(b)** A soccer player has a 70% success rate in penalty kicks. He takes 10 shots, and the number of goals is counted.
3. **(c)** A machine produces bolts. It is known that 2% of them have a flaw. From 1,000 bolts, 30 are inspected, and the number of flawed bolts is counted.
4. **(d)** In Germany, 80% of households have internet access. 1,000 households are randomly surveyed to determine how many have internet access.

### Explanation:
Each scenario describes a **binomial distribution**, which is a specific Bernoulli process where the probability of success \(p\) is known, and the process is repeated \(n\) times.

#### **General Formula for Binomial Distribution:**
\[
P(X = k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n-k}
\]
Where:
- \(n\) is the number of trials,
- \(k\) is the number of successes,
- \(p\) is the probability of success,
- \(\binom{n}{k}\) is the binomial coefficient.

### Solutions:
- **(a)**: Probability \(p = 0.5\) (fair coin), \(n = 5\) (5 flips).
- **(b)**: Probability \(p = 0.7\) (70% success rate), \(n = 10\) (10 penalty shots).
- **(c)**: Probability \(p = 0.02\) (2% defect rate), \(n = 30\) (30 bolts tested).
- **(d)**: Probability \(p = 0.8\) (80% internet access), \(n = 1,000\) (1,000 households surveyed).

Would you like to calculate specific probabilities or expected values for these cases?

### Relative Questions:
1. How is the binomial distribution applied in real-world situations?
2. How do you calculate the expected value in a binomial distribution?
3. What is the relationship between Bernoulli and binomial processes?
4. How do you calculate the variance of a binomial distribution?
5. How does sample size affect the accuracy of binomial probability estimates?

**Tip**: The expected value \(E(X)\) in a binomial distribution is simply \(n \cdot p\), where \(n\) is the number of trials and \(p\) is the probability of success.

Geben Sie die Länge und die Trefferwahrscheinlichkeit der folgenden Bernoulli-Ketten an: 
1. Eine Münze wird fünfmal geworfen. Es wird gezählt, wie oft Kopf fällt.
2. Ein Fußballspieler hat beim Elfmeterschießen eine Trefferquote von 70%. Er schießt zehn Elfmeter und dabei wird gezählt, wie oft er trifft.
3. Eine Maschine stellt Muttern her. Erfahrungsgemäß haben 2% einen Grat. Aus 1000 Muttern werden 30 entnommen und es wird gezählt, wie viele einen Grat haben.
4. In Deutschland verfügen ca. 80% der Haushalte über einen Internetzugang. Es werden 1000 Haushalte ausgewählt und befragt, ob sie einen Internetzugang haben.

Learn about Bernoulli chains and their applications in real-world scenarios: coin tosses, penalty kicks with a 70% success rate, quality control with 2% defect rates, and surveys of internet access. This example covers the binomial distribution formula and expected value.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation