Problem 23 - Algebra 2

Question

A spinner has 4 sections labeled \(A, B, C,\) and \(D .\) Can the spinner be designed so \(P(A)=\frac{1}{12}, P(B)=\frac{1}{6}, P(C)=\frac{1}{3},\) and \(P(D)=\frac{5}{12} ?\) If so, explain how.

Step-by-Step Solution

Answer

The sum of the probabilities \(P(A), P(B), P(C), P(D)\) is \(\frac{1}{12} + \frac{1}{6} + \frac{1}{3} + \frac{5}{12} = 1\). Therefore, it is possible to design the spinner with these probabilities for sections A, B, C, and D respectively.

Step 1: Understand the total probability rule

In any random experiment (like spinning a spinner), the combined probability of all possible outcomes must be 1. This is a fundamental concept in probability, known as the total probability rule.

Step 2: Add up the given probabilities

You need to add up the given probabilities \(P(A) = \frac{1}{12}, P(B) = \frac{1}{6}, P(C) = \frac{1}{3}, P(D) = \frac{5}{12}\). So the total probability \(P(Total) = P(A) + P(B) + P(C) + P(D)\).

Step 3: Check if sum of probabilities is 1

Now, you evaluate whether the total probability equals to 1. If \(P(Total) = 1\), then it means that the spinner can be designed as per the given probabilities. If not, then it's impossible to design such a spinner based on the provided probabilities.

Key Concepts

Total Probability Rule

The Total Probability Rule is a key principle in probability theory. It states that the sum of the probabilities of all possible outcomes of a random experiment is always equal to 1. This rule is essential to ensure that all potential scenarios are accounted for, and no possibility is left unaddressed.

For instance, if you spin a spinner divided into sections, each section will account for a part of that total probability. Together, all probabilities must add up to 1, reflecting the fact that when you spin the spinner, one of the outcomes must occur.

By understanding the Total Probability Rule, we ensure that probability values are correctly assigned and verified for whatever random experiment we are considering.

Random Experiment

A random experiment is any process or action where the outcome is uncertain. In the context of the probability problem we're discussing, spinning a spinner is an example of a random experiment.

Such experiments have several possible outcomes, which cannot be predicted with absolute certainty before the experiment is conducted. Each outcome has a probability associated with it, representing the chance of that specific result occurring.

Fundamentally, random experiments form the basis of probability studies, allowing us to predict likelihoods and make informed decisions based on potential results. Analyzing a spinner helps to visualize these concepts, turning abstract principles into something tangible.

Probability Sum

When dealing with probabilities, the concept of probability sum helps us check the validity of a probability distribution. To ensure a correct setup of a random experiment, like our spinner example, the sum of all individual outcome probabilities must equal 1.

Take the spinner problem: you're provided with probabilities for each section— \(P(A) = \frac{1}{12}\), \(P(B) = \frac{1}{6}\), \(P(C) = \frac{1}{3}\), and \(P(D) = \frac{5}{12}\). Adding these, we get:

• \(P(Total) = \frac{1}{12} + \frac{1}{6} + \frac{1}{3} + \frac{5}{12}\)

Converting each fraction to a common denominator and summing, the total should exactly equate to 1. If this is the case, then the probability distribution is valid.

Spinner Design

When designing a spinner, the probability of landing on each section needs to be considered carefully. To fit the given probabilities for sections A, B, C, and D, the spinner must be constructed such that these probabilities are accurately represented.

Consider how dividing the spinner should both reflect and ensure the calculated probabilities. Each probability like \(\frac{1}{12}\), \(\frac{1}{6}\), \(\frac{1}{3}\), and \(\frac{5}{12}\) determines relative section sizes on the spinner. Suppose the circumference of the spinner is seen as 1 whole unit; then each segment's arc length should be proportionate to its assigned probability.

Creating a spinner based on such clear probability guidelines helps to visualize probability concepts and ensures all outcomes are equally and fairly assessed.