Problem 10 - Algebra and Trigonometry

Question

Determine whether the following is a probability model. $$\begin{array}{lc}\text { Outcome } & \text { Probability } \\\\\hline \text { Erica } & 0.3 \\\\\text { Joanne } & 0.2 \\\\\text { Laura } & 0.1 \\\\\text { Donna } & 0.5 \\\\\text { Angela } & -0.1\end{array}$$

Step-by-Step Solution

Answer

No, it is not a valid probability model due to the negative probability for Angela.

Step 1 - Identify the conditions for a probability model

A probability model must satisfy two conditions: (1) Each individual probability must be between 0 and 1 (inclusive). (2) The sum of all probabilities must equal 1.

Step 2 - Check individual probabilities

Examine each probability: - Erica: 0.3 (valid)- Joanne: 0.2 (valid)- Laura: 0.1 (valid)- Donna: 0.5 (valid)- Angela: -0.1 (invalid)

Step 3 - Verify sum of probabilities

Add all the probabilities: 0.3 + 0.2 + 0.1 + 0.5 + (-0.1) = 1.0.The sum of all probabilities is 1 which satisfies the second condition.

Step 4 - Assess the validity of the model

Even though the sum of probabilities equals 1, the presence of a negative probability (-0.1 for Angela) means the first condition is violated. Therefore, this is not a valid probability model.

Key Concepts

probability conditions

In probability theory, a probability model sets up the framework to evaluate the likelihood of different outcomes. It must satisfy two essential conditions to be valid. First, each individual probability assigned to an outcome must lie between 0 and 1 (inclusive). This means any probability must be either a number within the segment between 0 and 1, inclusive of the endpoints, which represent the extremes of impossible (0) and certain (1) events respectively.
Secondly, the sum of all the probabilities of the individual outcomes must be exactly 1. This condition ensures that the model fully accounts for all possible outcomes, with no loss or excess in the total probability. Only if both these conditions are met can a set of probabilities be considered a valid probability model.

individual probabilities

Each individual probability reflects the chance of a specific outcome occurring. When we check these individual probabilities, each must satisfy the rule of being between 0 and 1 (inclusive).
Looking at the given exercise:

• Erica: 0.3
• Joanne: 0.2
• Laura: 0.1
• Donna: 0.5
• Angela: -0.1

Most values meet the condition. However, Angela has a probability of -0.1, which violates the first condition because probabilities cannot be negative. Thus, every outcome in a valid probability model must have a non-negative probability that does not exceed 1. By catching such discrepancies, we ensure the integrity of our probability model.

sum of probabilities

In addition to each individual probability being valid, the sum of all the probabilities in the model must equal 1. Summing up the probabilities in the exercise gives us:
\[0.3 + 0.2 + 0.1 + 0.5 + (-0.1) = 1.0\]
This sum indicates that the total probability is properly accounted for, so the second condition is satisfied. Nonetheless, even though the sum is correct, the model fails the first condition because of a negative individual probability. This underlines the importance of evaluating both conditions together. A valid probability model necessitates each individual probability to be within the range of [0,1] and that all probabilities combined should equal exactly 1 to ensure a full and accurate representation of all potential outcomes.