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Chapter 5: Problem 9

Why is the following not a probability model? $$ \begin{array}{lc} \text { Color } & \text { Probability } \\ \hline \text { Red } & 0.3 \\ \hline \text { Green } & -0.3 \\ \hline \text { Blue } & 0.2 \\ \hline \text { Brown } & 0.4 \\ \hline \text { Yellow } & 0.2 \\ \hline \text { Orange } & 0.2 \\ \hline \end{array} $$

Short Answer

Expert verified
The model is invalid because the probability of Green is negative, violating the rule that probabilities must be between 0 and 1.

Step by step solution

01

Understanding the Probability Model

A probability model must satisfy two conditions: all probabilities must be between 0 and 1, and the sum of all probabilities must be equal to 1.
02

Check Individual Probabilities

Examine each probability listed: Red (0.3), Green (-0.3), Blue (0.2), Brown (0.4), Yellow (0.2), Orange (0.2). Note that one of these probabilities is negative: Green (-0.3). Probabilities cannot be negative.
03

Sum the Probabilities

Calculate the sum of all probabilities: 0.3 + (-0.3) + 0.2 + 0.4 + 0.2 + 0.2. This sum is equal to 1.0.
04

Identify the Issue

Although the sum of the probabilities is 1, one of the probabilities is negative, which violates the rule that all probabilities must be between 0 and 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Rules
When dealing with probabilities, there are key rules that must always be followed. These rules help ensure that your probability model is correct and meaningful. One of the most important rules is that every probability value must fall between 0 and 1. This range includes 0 and 1 but does not allow for any negative probabilities or probabilities greater than 1.

If any probability falls outside this range, it compromises the integrity of your model. Also, a probability of 0 means the event is impossible, while a probability of 1 means the event is certain. Understanding these fundamental rules is crucial for creating any valid probability model and for solving related exercises correctly.
Sum of Probabilities
In any valid probability model, the sum of all individual probabilities must add up to exactly 1. This rule ensures that the model exhaustively covers all possible outcomes without any gaps or overlaps. When the probabilities are correctly summed up to 1, it confirms that you have accounted for every possible event in your scenario.

For example, let's take a simple probability model with outcomes like flipping a coin. If the probability of landing heads is 0.5 and the probability of landing tails is also 0.5, the sum of these probabilities is 1 (0.5 + 0.5 = 1).

Failing to meet this sum criterion indicates that the probability model is either incomplete or incorrect. Therefore, always double-check the total sum to make sure it equals 1.
Valid Probability Range
The validity of each probability in a model is determined by its range. As mentioned before, every probability should be between 0 and 1. For example, if an event has a probability of 0.3, it means there's a 30% chance that event will occur. If the probability is 0.7, it means there's a 70% chance.

In our example exercise, one of the probabilities listed is -0.3. This is not valid because probabilities cannot be negative. Negative probabilities do not make sense as they imply negative chances, which are logically impossible.

Remember, always double-check each value in your probability model to ensure every single one is between 0 and 1. This rule is essential for maintaining an accurate and valid probability model.

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Most popular questions from this chapter

The grade appeal process at a university requires that a jury be structured by selecting five individuals randomly from a pool of eight students and ten faculty. (a) What is the probability of selecting a jury of all students? (b) What is the probability of selecting a jury of all faculty? (c) What is the probability of selecting a jury of two students and three faculty?

A box containing twelve 40-watt light bulbs and eighteen 60-watt light bulbs is stored in your basement. Unfortunately, the box is stored in the dark and you need two 60-watt bulbs. What is the probability of randomly selecting two 60-watt bulbs from the box?

A certain digital music player randomly plays each of 10 songs. Once a song is played, it is not repeated until all the songs have been played. In how many different ways can the player play the 10 songs?

Suppose that you just received a shipment of six televisions and two are defective. If two televisions are randomly selected, compute the probability that both televisions work. What is the probability that at least one does not work?

Adult Americans (18 years or older) were asked whether they used social media (Facebook, Twitter, and so on ) regularly. The following table is based on the results of the survey. $$ \begin{array}{lccccc} & \mathbf{1 8 - 3 4} & \mathbf{3 5 - 4 4} & \mathbf{4 5 - 5 4} & \mathbf{5 5 +} & \text { Total } \\ \hline \begin{array}{l} \text { Use social } \\ \text { media } \end{array} & 117 & 89 & 83 & 49 & \mathbf{3 3 8} \\ \hline \begin{array}{l} \text { Do not use } \\ \text { social media } \end{array} & 33 & 36 & 57 & 66 & \mathbf{1 9 2} \\ \hline \text { Total } & \mathbf{1 5 0} & \mathbf{1 2 5} & \mathbf{1 4 0} & \mathbf{1 1 5} & \mathbf{5 3 0} \\ \hline \end{array} $$ (a) What is the probability that a randomly selected adult American uses social media, given the individual is \(18-34\) years of age? (b) What is the probability that a randomly selected adult American is \(18-34\) years of age, given the individual uses social media? (c) Are 18 - to 34 -year olds more likely to use social media than individuals in general? Why?
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