In Exercises 1–4, find the indicated probability using the geometric distribution.


Find P(5) when p = 0.09

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, p is the probability of success, and (1 - p) is the probability of failure.
2. Step 2: Identify the given values in the problem. Here, k = 5 (the trial number) and p = 0.09 (the probability of success).
3. Step 3: Substitute the given values into the formula. This becomes P(5) = (1 - 0.09)^(5 - 1) * 0.09.
4. Step 4: Simplify the expression. First, calculate (1 - 0.09) to find the probability of failure, then raise it to the power of (5 - 1), and finally multiply by 0.09.
5. Step 5: After simplifying the expression, you will have the probability P(5). Ensure all calculations are performed accurately to find the final result.

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 constant probability of success, denoted as 'p'. 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) provides the probabilities of discrete random variables. For the geometric distribution, the PMF calculates the likelihood of achieving the first success on the k-th trial. Understanding the PMF is essential for determining specific probabilities, such as P(5) in this case, which represents the probability that the first success occurs on the fifth trial.

Calculating Geometric Probability

To calculate the probability of the first success occurring on the k-th trial using the geometric distribution, substitute the values of p and k into the PMF formula. For example, with p = 0.09 and k = 5, the calculation involves finding (1 - 0.09)^(5-1) * 0.09. This process illustrates how to apply the geometric distribution to find specific probabilities.