Problem 8 - Algebra and Trigonometry

Question

Determine whether the following is a probability model. $$\begin{array}{lc}\text { Outcome } & \text { Probability } \\\\\hline \text { Steve } & 0.4 \\\\\text { Bob } & 0.3 \\ \text { Faye } & 0.1 \\\\\text { Patricia } & 0.2\end{array}$$

Step-by-Step Solution

Answer

Yes, it is a probability model.

Step 1: Identify the Probabilities

List the probabilities given for each outcome: - Steve: 0.4 - Bob: 0.3 - Faye: 0.1 - Patricia: 0.2

Step 2: Check if Probabilities are Between 0 and 1

Ensure that each probability is between 0 and 1: - 0.4 (Steve) is between 0 and 1 - 0.3 (Bob) is between 0 and 1 - 0.1 (Faye) is between 0 and 1 - 0.2 (Patricia) is between 0 and 1 Since all probabilities are between 0 and 1, move to the next step.

Step 3: Sum the Probabilities

Add the probabilities together: data to be processed- 0.4 (Steve) + 0.3 (Bob) + 0.1 (Faye) + 0.2 (Patricia) = 1.0

Step 4: Confirm the Sum is 1

The probability sum is 1.0, which satisfies the condition for a probability model.

Key Concepts

probabilities

Probabilities are a fundamental concept in any probability model. They represent the likelihood of different outcomes.
A probability is a number between 0 and 1, where 0 means the event never occurs and 1 means the event always occurs.
Every possible outcome of an experiment is assigned a probability. For example, in this exercise, we have these probabilities:

• Steve: 0.4
• Bob: 0.3
• Faye: 0.1
• Patricia: 0.2

Each of these numbers tells us how likely each person is to be chosen.
Checking if each probability is within the range of 0 and 1 is crucial before combining them to form a proper probability model.

sum of probabilities

One of the key properties of a valid probability model is that the sum of all probabilities should equal 1.
This means that the combined probability of all possible outcomes must account for 100% of the events.

The process involves adding the probabilities of individual outcomes. For this exercise, we add:
\( 0.4 \, (Steve) \ + \, 0.3 \, (Bob) \ + \, 0.1 \, (Faye) \ + \, 0.2 \, (Patricia) = 1.0 \)
This confirms that we have a complete probability model, as the total reaches exactly 1.0.
If the sum had not been 1.0, our model would be incorrect and would need adjusting.

checking probability conditions

To determine if a set of probabilities forms a valid probability model, two main conditions must be checked:
1. Each probability is between 0 and 1
2. The sum of all probabilities is 1
In this exercise, first, we checked that:
\( 0.4 \, (Steve) \ \in \ [0,1] , \ 0.3 \, (Bob) \ \in \ [0,1] , \ 0.1 \, (Faye) \ \in \ [0,1] , \ 0.2 \, (Patricia) \ \in \ [0,1] \)

Next, we verified that the sum of these probabilities is 1.0:
\( 0.4 + 0.3 + 0.1 + 0.2 = 1.0 \)
This ensures that our probability model fulfills the necessary conditions and is therefore valid.
Always remember these steps when trying to verify any probability model.