Assume that a procedure yields a binomial distribution with a trial repeated n=5n=5 times. Use some form of technology to find the probability distribution given the probability p=0.687 of success on a single trial.
(s + f)^5
s^5 + 5 s^4 f + 10 s^3 f^2 +
... 10 s^2 f^3 + 5 s f^4 + f^5
s = p(success) = .687
f = p(failure) = 1 - p(success) = .313
To find the probability distribution for a binomial distribution with a given probability of success, you can use a calculator or statistical software. Here's how you can do it using a statistical software like Python:
Step 1: Install Python and the SciPy library if you haven't already. You can install Python from python.org, and install SciPy by running the command "pip install scipy" in your command prompt or terminal.
Step 2: Open a Python script or interactive environment (e.g., Jupyter Notebook) and import the necessary libraries:
```python
from scipy.stats import binom
```
Step 3: Define the number of trials (n) and the probability of success (p):
```python
n = 5
p = 0.687
```
Step 4: Use the binom.pmf() function to calculate the probability mass function (PMF) for each possible number of successes:
```python
prob_distribution = [binom.pmf(k, n, p) for k in range(n+1)]
```
The binom.pmf() function calculates the probability of having exactly k successes in n trials, given the probability of success p.
Step 5: Print the probability distribution:
```python
for k, prob in enumerate(prob_distribution):
print(f'P(X={k}) = {prob}')
```
The above code will display the probability for each possible value of X (number of successes).
By following these steps and executing the code, you will obtain the probability distribution for the given binomial distribution with n=5 and p=0.687.