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Probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. In our case, we are interested in a special type of probability distribution known as the geometric distribution. This particular distribution models scenarios where you repeat a process or "experiment" until you achieve a specific result, called a "success."

For example, in the village of Nanakuli, the experiment is meeting the residents until you find the first person of Hawaiian ancestry. Each resident you meet can be either a success (of Hawaiian ancestry) or a failure (not of Hawaiian ancestry). The geometric distribution helps us to find out how many people we need to meet before experiencing our first "success."

• In Nanakuli, the probability of success (finding a person of Hawaiian ancestry) is 0.8. This is incorporated into the formula for the geometric distribution, which is: \[ P(n) = (1-p)^{n-1} \cdot p \]
• For Waikiki, the probability of success is much lower, at 0.04. The same formula applies but results in a different probability distribution for the number of people you expect to meet before a success is achieved.

A random variable is a variable whose value depends on the outcomes of a random phenomenon. It essentially provides a bridge between random phenomena and real numbers. In the context of probability and statistics, random variables can take on various values with a specific probability attached to each.

In this exercise, the random variable, denoted by \( n \), represents the number of people you need to meet until you encounter the first person of Hawaiian ancestry. In other words, \( n \) captures the random nature of needing a variable number of attempts to find a success.

• The random variable can take any positive integer value, such as 1, 2, 3, and so on.
• A small value of \( n \) indicates an encounter with a person of Hawaiian ancestry quickly, while a larger \( n \) suggests you would encounter more non-Hawaiian individuals before a Hawaiian individual.
• This encapsulation of randomness into a variable allows us to apply mathematical formulas to predict and analyze real-world phenomena.

The probability of success is a crucial component in understanding and calculating probability distributions, especially in scenarios modeled by the geometric distribution. It refers to the likelihood of achieving a specific desired outcome in a single trial.

In our investigation of Nanakuli and Waikiki, the probability of success is represented by the proportion of residents of Hawaiian ancestry. This probabiliy is defined as \( p \):

• For Nanakuli, where 80% of the residents are of Hawaiian ancestry, \( p = 0.8 \).
• In contrast, Waikiki has a much smaller percentage of Hawaiian ancestry, with \( p = 0.04 \).

The probability of success directly influences the shape and characteristics of the geometric distribution. A higher probability of success means fewer trials are needed on average to achieve a success, while a lower probability requires more trials. This probability shapes expectations about how many people you'd need to meet before encountering someone of Hawaiian ancestry. Understanding this concept is essential when applying the geometric distribution formula to real-world situations.