Question

assume the geometric distribution applies. use the given probability of success p to find the indicated probability. find p(5) when p = 0.10. p(5) = \boxed{} (round to five decimal places as needed.)

Explanation:

Step1: Recall Geometric Probability Formula

The formula for the probability that the first success occurs on the \( n \)-th trial in a geometric distribution is \( P(n) = (1 - p)^{n - 1} \cdot p \), where \( p \) is the probability of success on a single trial.

Step2: Identify Values

Here, \( n = 5 \) and \( p = 0.10 \). First, calculate \( 1 - p \): \( 1 - 0.10 = 0.90 \).

Step3: Substitute into Formula

Substitute \( n = 5 \), \( p = 0.10 \), and \( 1 - p = 0.90 \) into the formula:
\( P(5) = (0.90)^{5 - 1} \cdot 0.10 \)
Simplify the exponent: \( 5 - 1 = 4 \), so \( P(5) = (0.90)^4 \cdot 0.10 \).

Step4: Calculate \( (0.90)^4 \)

\( (0.90)^4 = 0.90 \times 0.90 \times 0.90 \times 0.90 = 0.6561 \).

Step5: Multiply by \( p \)

Multiply \( 0.6561 \) by \( 0.10 \): \( 0.6561 \times 0.10 = 0.06561 \).

Answer:

\( 0.06561 \)