Question

assume the geometric distribution applies. use the given probability of success p to find the indicated probability. find p(4) when p = 0.80. p(4) = \boxed{} (round to five decimal places as needed.)

Explanation:

Step1: Recall Geometric Probability Formula

The formula for the geometric distribution is \( P(X = k)=(1 - p)^{k - 1}\cdot p \), where \( k \) is the number of trials until the first success, and \( p \) is the probability of success on a single trial.

Step2: Identify Values of \( k \) and \( p \)

Here, \( k = 4 \) and \( p = 0.80 \). So, \( 1 - p=1 - 0.80 = 0.20 \).

Step3: Substitute Values into Formula

Substitute \( k = 4 \), \( p = 0.80 \), and \( 1 - p = 0.20 \) into the formula:
\( P(4)=(0.20)^{4 - 1}\cdot0.80=(0.20)^{3}\cdot0.80 \)

Step4: Calculate the Result

First, calculate \( (0.20)^{3}=0.20\times0.20\times0.20 = 0.008 \).
Then, multiply by \( 0.80 \): \( 0.008\times0.80 = 0.0064 \).

Answer:

\( 0.00640 \) (rounded to five decimal places)