Suppose A Large Technology Company Has 53% Of Its Workforce Comprised Of Software Developers. A Committee Of 10 Employees Is Chosen At Random From The Entire Staff. Let { Y $}$ Denote The Number Of Software Developers Selected For The
Understanding the Problem
In this scenario, we are dealing with a large technology company that has a significant number of software developers in its workforce. The company has a total of 100 employees, out of which 53% are software developers. This means that there are 53 software developers and 47 non-software developers in the company.
Defining the Random Variable
Let { Y $}$ denote the number of software developers selected for the committee of 10 employees. This random variable represents the number of software developers chosen from the total of 100 employees.
Calculating the Probability Distribution
To calculate the probability distribution of { Y $}$, we need to find the probability of selecting k software developers out of 10 employees, where k can take values from 0 to 10.
Using the Hypergeometric Distribution
The probability distribution of { Y $}$ can be modeled using the hypergeometric distribution. The hypergeometric distribution is a discrete probability distribution that models the number of successes (in this case, software developers) in a fixed-size sample (the committee of 10 employees) drawn without replacement from a finite population (the total of 100 employees).
Calculating the Probability Mass Function
The probability mass function (PMF) of the hypergeometric distribution is given by:
P(Y = k) = \frac{\binom{53}{k} \binom{47}{10-k}}{\binom{100}{10}}
where \binom{n}{r} is the binomial coefficient, which represents the number of ways to choose r items from a set of n items.
Calculating the Expected Value and Variance
The expected value (E[Y]) and variance (Var[Y]) of the hypergeometric distribution can be calculated using the following formulas:
E[Y] = n * p Var[Y] = n * p * (1-p) * (N-n) / (N-1)
where n is the sample size (10), p is the proportion of software developers in the population (53/100), and N is the total population size (100).
Interpreting the Results
The expected value of Y represents the average number of software developers selected for the committee, while the variance represents the spread of the distribution. The probability mass function can be used to calculate the probability of selecting a specific number of software developers for the committee.
Example Calculations
Let's calculate the probability of selecting 5 software developers out of 10 employees using the hypergeometric distribution.
P(Y = 5) = \frac{\binom{53}{5} \binom{47}{5}}{\binom{100}{10}} ≈ 0.23
This means that there is approximately a 23% chance of selecting 5 software developers out of 10 employees.
Conclusion
In this scenario, we used the hypergeometric distribution to model the number of software developers selected for a committee of 10 employees. We calculated the probability mass function, expected value, and variance of the distribution, and interpreted the results. The hypergeometric distribution provides a useful tool for modeling the number of successes in a fixed-size sample drawn without replacement from a finite population.
Further Reading
For further reading on the hypergeometric distribution, we recommend the following resources:
- Hypergeometric Distribution by Wolfram MathWorld
- Hypergeometric Distribution by Stat Trek
- Hypergeometric Distribution by Math Is Fun
References
- Hypergeometric Distribution by Johnson, N. L., Kotz, S., & Kemp, A. W. (1992). Univariate discrete distributions (2nd ed.). Wiley.
- Hypergeometric Distribution by Hogg, R. V., & Tanis, E. A. (2001). Probability and statistical inference (6th ed.). Prentice Hall.
Glossary
- Hypergeometric Distribution: A discrete probability distribution that models the number of successes in a fixed-size sample drawn without replacement from a finite population.
- Probability Mass Function: A function that assigns a probability to each possible value of a random variable.
- Expected Value: The average value of a random variable.
- Variance: A measure of the spread of a distribution.
Understanding the Hypergeometric Distribution
The hypergeometric distribution is a discrete probability distribution that models the number of successes in a fixed-size sample drawn without replacement from a finite population. It is commonly used in statistics and probability theory to analyze data from experiments, surveys, and other studies.
Frequently Asked Questions
Q: What is the hypergeometric distribution?
A: The hypergeometric distribution is a discrete probability distribution that models the number of successes in a fixed-size sample drawn without replacement from a finite population.
Q: What are the key parameters of the hypergeometric distribution?
A: The key parameters of the hypergeometric distribution are:
- N: The total population size
- n: The sample size
- K: The number of successes in the population
- k: The number of successes in the sample
Q: How is the hypergeometric distribution used in real-world applications?
A: The hypergeometric distribution is used in a variety of real-world applications, including:
- Quality control: To determine the probability of a certain number of defective products in a sample
- Medical research: To analyze the probability of a certain number of patients with a specific disease in a sample
- Marketing research: To determine the probability of a certain number of customers with a specific characteristic in a sample
Q: What are the assumptions of the hypergeometric distribution?
A: The assumptions of the hypergeometric distribution are:
- Random sampling: The sample is drawn randomly from the population
- No replacement: The sample is drawn without replacement from the population
- Finite population: The population is finite and known
Q: How is the hypergeometric distribution calculated?
A: The hypergeometric distribution is calculated using the following formula:
P(X = k) = \frac{\binom{K}{k} \binom{N-K}{n-k}}{\binom{N}{n}}
where \binom{n}{r} is the binomial coefficient.
Q: What are the advantages and disadvantages of the hypergeometric distribution?
A: The advantages of the hypergeometric distribution are:
- Accurate modeling: The hypergeometric distribution accurately models the number of successes in a fixed-size sample drawn without replacement from a finite population
- Flexibility: The hypergeometric distribution can be used to model a variety of real-world applications
The disadvantages of the hypergeometric distribution are:
- Complexity: The hypergeometric distribution can be complex to calculate and interpret
- Assumptions: The hypergeometric distribution assumes random sampling, no replacement, and a finite population, which may not always be the case in real-world applications
Q: What are some common mistakes to avoid when using the hypergeometric distribution?
A: Some common mistakes to avoid when using the hypergeometric distribution are:
- Incorrect assumptions: Failing to meet the assumptions of the hypergeometric distribution, such as random sampling and no replacement
- Incorrect calculation: Failing to calculate the hypergeometric distribution correctly
- Incorrect interpretation: Failing to interpret the results of the hypergeometric distribution correctly
Conclusion
The hypergeometric distribution is a powerful tool for modeling the number of successes in a fixed-size sample drawn without replacement from a finite population. By understanding the key parameters, assumptions, and calculations of the hypergeometric distribution, you can accurately model and analyze real-world data.
Further Reading
For further reading on the hypergeometric distribution, we recommend the following resources:
- Hypergeometric Distribution by Wolfram MathWorld
- Hypergeometric Distribution by Stat Trek
- Hypergeometric Distribution by Math Is Fun
References
- Hypergeometric Distribution by Johnson, N. L., Kotz, S., & Kemp, A. W. (1992). Univariate discrete distributions (2nd ed.). Wiley.
- Hypergeometric Distribution by Hogg, R. V., & Tanis, E. A. (2001). Probability and statistical inference (6th ed.). Prentice Hall.
Glossary
- Hypergeometric Distribution: A discrete probability distribution that models the number of successes in a fixed-size sample drawn without replacement from a finite population.
- Probability Mass Function: A function that assigns a probability to each possible value of a random variable.
- Expected Value: The average value of a random variable.
- Variance: A measure of the spread of a distribution.