Questions: Assume the geometric distribution applies. Use the given probability of success p to find the indicated probability. Find P(3) when p=0.80. P(3) = (Round to five decimal places as needed.)

Solution Steps

We are given the probability of success \( p = 0.80 \) and we need to find \( P(3) \), which represents the probability that the first success occurs on the 3rd trial.

The probability mass function for a geometric distribution is given by:

\[ P(k) = (1 - p)^{k-1} \cdot p \]

Substituting the values:

\[ P(3) = (1 - 0.80)^{3-1} \cdot 0.80 \]

Calculating the components:

\[ 1 - p = 1 - 0.80 = 0.20 \]

Now substituting back into the formula:

\[ P(3) = (0.20)^{2} \cdot 0.80 \]

Calculating \( (0.20)^{2} \):

\[ (0.20)^{2} = 0.04 \]

Now, substituting this value:

\[ P(3) = 0.04 \cdot 0.80 = 0.032 \]

Final Answer