Questions: A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n=10, p=0.9, x=8 P(8)= (Do not round until the final answer. Then round to four decimal places as needed.)

Solution Steps

We are tasked with calculating the probability of obtaining \( x = 8 \) successes in \( n = 10 \) independent trials of a binomial experiment, where the probability of success on each trial is \( p = 0.9 \).

The probability of exactly \( x \) successes in \( n \) trials is given by the formula:

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

where \( q = 1 - p \) is the probability of failure. For our parameters:

• \( n = 10 \)
• \( x = 8 \)
• \( p = 0.9 \)
• \( q = 1 - 0.9 = 0.1 \)

The binomial coefficient \( \binom{n}{x} \) is calculated as follows:

\[ \binom{10}{8} = \frac{10!}{8!(10-8)!} = \frac{10 \times 9}{2 \times 1} = 45 \]

Now we can substitute the values into the probability formula:

\[ P(X = 8) = 45 \cdot (0.9)^8 \cdot (0.1)^{10-8} \]

Calculating \( (0.9)^8 \) and \( (0.1)^2 \):

\[ (0.9)^8 \approx 0.43046721 \quad \text{and} \quad (0.1)^2 = 0.01 \]

Now substituting these values back into the formula:

\[ P(X = 8) = 45 \cdot 0.43046721 \cdot 0.01 \approx 0.1937 \]

Final Answer