Assume that a procedure yields a binomial distribution with a trial repeated n=5 
 times. Use some form of technology to find the probability distribution given the probability p=0.528  
 of success on a single trial.

(Report answers accurate to 4 decimal places.)

The binomial probability distribution for \( n = 5 \) trials and a probability of success \( p = 0.528 \) on a single trial is as follows:

- \( P(X = 0) = 0.0234 \)
- \( P(X = 1) = 0.1310 \)
- \( P(X = 2) = 0.2932 \)
- \( P(X = 3) = 0.3279 \)
- \( P(X = 4) = 0.1834 \)
- \( P(X = 5) = 0.0410 \)

These probabilities sum to 1, confirming the validity of the distribution.

Would you like more details or have any questions?

Here are some follow-up questions to explore further:
1. What is the expected value of the binomial distribution in this case?
2. How would you compute the variance and standard deviation of this distribution?
3. What is the cumulative probability of getting at most 3 successes?
4. How does changing \( p \) to a lower or higher value affect the distribution shape?
5. Can you derive the moment-generating function for a binomial distribution?

**Tip:** In binomial distributions, the probability of success \( p \) and the number of trials \( n \) significantly influence the skewness of the distribution.

Assume that a procedure yields a binomial distribution with a trial repeated n=5 times. Use some form of technology to find the probability distribution given the probability p=0.528 of success on a single trial. (Report answers accurate to 4 decimal places.)

Explore how to calculate the binomial probability distribution for 5 trials and a probability of 0.528 of success per trial. Detailed step-by-step solution with answers accurate to 4 decimal places, covering key concepts like binomial distribution and probability. Suitable for high school students in Grades 11-12.

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