In Exercises 1–4, find the indicated probability using the geometric distribution.
Find P(3) when p = 0.65

Verified step by step guidance

1. Step 1: Recall the formula for the probability mass function (PMF) of a geometric distribution: P(X = k) = (1 - p)^(k - 1) * p, where k is the trial number, and p is the probability of success on a single trial.
2. Step 2: Identify the given values from the problem. Here, k = 3 (the third trial), and p = 0.65 (the probability of success).
3. Step 3: Substitute the given values into the formula. Replace k with 3 and p with 0.65 in the formula: P(3) = (1 - 0.65)^(3 - 1) * 0.65.
4. Step 4: Simplify the expression. First, calculate (1 - 0.65), which represents the probability of failure. Then raise this value to the power of (3 - 1), which is 2. Finally, multiply the result by 0.65.
5. Step 5: The result of the above calculation will give you the probability P(3). Ensure all calculations are performed accurately to find the final value.

Key Concepts

Geometric Distribution

The geometric distribution models the number of trials needed to achieve the first success in a series of independent Bernoulli trials. It is characterized by a single parameter, p, which represents the probability of success on each trial. The probability mass function is given by P(X = k) = (1 - p)^(k-1) * p, where k is the trial number on which the first success occurs.

Probability Mass Function (PMF)

The probability mass function (PMF) of a discrete random variable provides the probabilities of each possible value that the variable can take. For the geometric distribution, the PMF allows us to calculate the probability of achieving the first success on the k-th trial. Understanding the PMF is essential for solving problems related to discrete distributions, including finding specific probabilities.

Calculating P(3) for Geometric Distribution

To find P(3) in the context of the geometric distribution with p = 0.65, we apply the PMF formula. Specifically, we calculate P(X = 3) = (1 - 0.65)^(3-1) * 0.65, which simplifies to (0.35)^2 * 0.65. This calculation illustrates how to determine the probability of the first success occurring on the third trial, emphasizing the application of the geometric distribution in practical scenarios.