Questions: When using the binomial probability formula, q represents the Choose... n represents the Choose... p represents the Choose... x represents the Choose...

Solution Steps

Using the binomial probability formula, we can calculate the probability of obtaining exactly \( x \) successes in \( n \) trials. The formula is given by:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

For our example, with \( n = 10 \), \( x = 5 \), \( p = 0.5 \), and \( q = 0.5 \), we find:

\[ P(X = 5) = \binom{10}{5} \cdot (0.5)^5 \cdot (0.5)^{10-5} = 0.2461 \]

The mean (expected value) of a binomial distribution is calculated using the formula:

\[ \mu = n \cdot p \]

Substituting our values:

\[ \mu = 10 \cdot 0.5 = 5.0 \]

The variance of a binomial distribution is given by:

\[ \sigma^2 = n \cdot p \cdot q \]

Using our values:

\[ \sigma^2 = 10 \cdot 0.5 \cdot 0.5 = 2.5 \]

The standard deviation is the square root of the variance:

\[ \sigma = \sqrt{n \cdot p \cdot q} = \sqrt{10 \cdot 0.5 \cdot 0.5} = 1.5811 \]

In the context of the binomial probability formula:

• \( q \) represents the probability of failure on each trial.
• \( n \) represents the number of trials.
• \( p \) represents the probability of success on each trial.
• \( x \) represents the number of successes.

Final Answer