Problem 37 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Bob, the proprietor of Midland Lumber, feels that the odds in favor of a business deal going through are 9 to 5\. What is the (subjective) probability that this deal will not materialize?

Step-by-Step Solution

Answer

The subjective probability that the deal will not materialize is \(\frac{5}{14}\).

Step 1: Understanding the problem

Bob believes that the odds in favor of a business deal going through are 9 to 5. Odds are a ratio between the number of ways something can happen (success) and the number of ways it can't happen (failure). So in this case, 9 represents the number of ways the deal can go through, and 5 represents the number of ways it can't.

Step 2: Find the probability of the deal materializing

To find the probability of the deal materializing, we can use the odds to calculate the ratio of successful outcomes to the total number of possible outcomes. The probability of the deal going through is given by: \(P(\text{deal materializing}) = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}\) The total number of possible outcomes is the sum of the successful outcomes (9) and unsuccessful outcomes (5). So we have: \(P(\text{deal materializing}) = \frac{9}{9+5}\)

Step 3: Calculate the probability of the deal materializing

Now, calculate the probability of the deal materializing: \(P(\text{deal materializing}) = \frac{9}{9+5} = \frac{9}{14}\)

Step 4: Calculate the probability of the deal not materializing

To find the probability of the deal not materializing, subtract the probability of the deal materializing from 1: \(P(\text{deal not materializing}) = 1 - P(\text{deal materializing})\)

Step 5: Calculate the probability of the deal not materializing

Plug in the probability of the deal materializing that we calculated in Step 3: \(P(\text{deal not materializing}) = 1 - \frac{9}{14}\)

Step 6: Simplify the probability of the deal not materializing

Simplify the fraction to get the final probability: \(P(\text{deal not materializing}) = \frac{14-9}{14} = \frac{5}{14}\) So, the subjective probability that this deal will not materialize is \(\frac{5}{14}\).

Key Concepts

Odds Ratio

The odds ratio is a key concept in probability and statistics, used to compare the relative odds of an event occurring under different circumstances. It's important, especially in fields like gambling and decision-making, to understand how odds are expressed. In the exercise we've got, the odds are presented as '9 to 5' in favor of a business deal going through.

This means there are 9 ways for success and 5 ways for failure. To relate this to probability, the 'odds in favor' tell us how much more likely the event is to happen than not. To break this down into a form that might be easier to utilize, we often convert these odds into probability. While the odds ratio is a comparative measure, the probability is an absolute measure that gives us a clearer picture of just how likely (or unlikely) an event might be.

Therefore, understanding odds ratio is crucial for interpreting risk, chances of success or failure, and making informed decisions based on those ratios.

Probability Calculations

Calculating probability is fundamental to understanding chance and making predictions about events. Probability calculations can be intuitive sometimes, but they always rely on a sound mathematical basis. In the context of our exercise, the calculation involves turning the given odds into a usable probability that describes the likelihood of the business deal not materializing.

To do this, we first figure out the total number of possible outcomes by adding the ways an event can occur and the ways it cannot. Then, we divide the number of ways the event can occur (or not occur) by this total to get the probability. Understanding this process is essential. It helps break down complex real-world situations into understandable figures, allowing us to make predictions and decisions based on quantitative data. Students should practice this skill to gain confidence in tackling a variety of problems and interpreting statistical data in meaningful ways.

Mathematical Probability

Mathematical probability is the measure of the likelihood that an event will occur. It ranges from 0 (the event will definitely not occur) to 1 (the event will definitely occur). Probability can be represented by simple fractions, percentages, or decimals.

In the given exercise, we find that the subjective probability that the business deal will not materialize is \(\frac{5}{14}\). To understand the significance, we can see that the probability is less than 1 but more than 0, indicating that the event is possible but not certain. It's subjects like these where mathematical probabilities shine, helping us quantify and communicate chance in a precise language. Always remember, probability is all about a stable and consistent approach to uncertainty—a concept that is as much philosophical as it is mathematical. A solid grasp of mathematical probability will therefore not only assist students with their homework but also with reasoning and decision-making in their daily lives.