Problem 64 - Calculus for Biology and Medicine

Question

A random experiment consists of rolling a fair die until the first time a five or a six appears. Find the probability that the first five or six appears on the \(k\) th trial for \(k=1,2, \ldots, 5\).

Step-by-Step Solution

Answer

The probabilities are: \( P(X=1) = \frac{1}{3} \), \( P(X=2) = \frac{2}{9} \), \( P(X=3) = \frac{4}{27} \), \( P(X=4) = \frac{8}{81} \), and \( P(X=5) = \frac{16}{243} \).

Step 1: Understanding the problem

We need to determine the probability that a five or a six appears for the first time on the \(k\) th roll of a die, where \(k\) can be 1, 2, 3, 4, or 5.

Step 2: Identify the Success and Failure Probability

Since a die has 6 faces, the probability of rolling a 5 or 6 is \( \frac{2}{6} = \frac{1}{3} \), which represents a success. Therefore, the probability of not rolling a 5 or 6 (failure) is \( 1 - \frac{1}{3} = \frac{2}{3} \).

Step 3: Define the Geometric Distribution

The experiment follows a geometric distribution because we're interested in the number of trials up to and including the first success (rolling a 5 or a 6). For a geometric distribution where success occurs on the \(k\)th trial, the probability is given by \( P(X = k) = p (1-p)^{k-1} \).

Step 4: Calculating the Probability for each \(k\)

For each \(k = 1, 2, 3, 4, 5\), apply the formula for geometric probability: \[ P(X = k) = \left( \frac{1}{3} \right) \left( \frac{2}{3} \right)^{k-1} \]. Calculate for \(k = 1\): \(\frac{1}{3}(\frac{2}{3})^0 = \frac{1}{3}\), for \(k = 2\): \(\frac{1}{3}(\frac{2}{3})^1 = \frac{2}{9}\), and so on until \(k = 5\).

Step 5: Conclusion

The probabilities for rolling a 5 or 6 for the first time on trials 1 through 5 are: \( P(X = 1) = \frac{1}{3}, \) \( P(X = 2) = \frac{2}{9} \), \( P(X = 3) = \frac{4}{27}\), \( P(X = 4) = \frac{8}{81} \), \( P(X = 5) = \frac{16}{243} \).

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that focuses on the analysis of random events. It provides a framework for quantifying uncertainty, allowing us to assign a numerical value to the likelihood of different outcomes. This is particularly used in scenarios where the outcomes are unpredictable and can vary when repeated under identical conditions.
In probability theory:

• Every event has an associated probability, which is a number between 0 and 1. A probability of 0 means the event will never occur, while a probability of 1 indicates certainty.
• The sum of probabilities of all possible outcomes of a random experiment is always equal to 1.
• Probability theory can be applied to simple experiments, like tossing a coin, as well as complex situations involving multiple variables.

Understanding these basics is crucial for delving into more specific topics, such as geometric distributions, which are a part of discrete probability distributions. They specifically deal with scenarios where we are interested in the number of trials needed to achieve the first success.

Random Experiments

A random experiment is an action or process that produces a set of possible outcomes, and each outcome is uncertain until it is observed. Each trial in a random experiment is independent, meaning the result of one trial does not affect the next.
In the context of rolling a die:

• Each roll of the die is a trial in our random experiment.
• The outcome is uncertain. Each trial can result in any one of the six numbers: 1, 2, 3, 4, 5, or 6.
• When understanding this problem, the random experiment specifically involves rolling the die until we get either a 5 or a 6. "Success" occurs when a 5 or 6 is rolled for the first time.

This setup is foundational for using probability distributions, like the geometric distribution, to predict and calculate the likelihood of outcomes, such as how many rolls it will take to achieve the first successful roll.

Geometric Probability Formula

The geometric probability formula is the foundation for calculating the probability of the first success occurring on the k-th trial in a series of independent trials. It's a type of discrete probability distribution used when looking at independent trials with only two possible outcomes: success or failure.
The formula itself is given by:\[P(X = k) = p (1-p)^{k-1}\]Where:

• \(P(X = k)\) represents the probability that the first success occurs on the k-th trial.
• \(p\) is the probability of success on each trial.
• \((1-p)\) is the probability of failure on each trial.
• \(k-1\) accounts for each trial needed before the first success is achieved.

In the example of rolling a die to get a 5 or a 6, the formula helps to calculate the likelihood that these numbers appear on a specific trial. This exercise demonstrates the application of the geometric distribution by using the probability of rolling a 5 or 6 on the first through fifth trials.