Problem 35 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Steffi feels that the odds in favor of her winning her tennis match tomorrow are 7 to \(5 .\) What is the (subjective) probability that she will win her match tomorrow?

Step-by-Step Solution

Answer

The subjective probability that Steffi will win her tennis match tomorrow is \(\frac{7}{12}\).

Step 1: Understanding the odds

Odds in favor of an event occurring are the ratio of the number of favorable outcomes to the number of unfavorable outcomes. In this case, the odds in favor of Steffi winning her tennis match tomorrow are 7 to 5. This means there are 7 favorable outcomes (Steffi wins) and 5 unfavorable outcomes (Steffi loses).

Step 2: Calculate the total number of outcomes

The total number of outcomes is the sum of favorable outcomes (winning) and unfavorable outcomes (losing). In this problem, the total number of outcomes is 7 (favorable) + 5 (unfavorable) = 12.

Step 3: Calculate the probability

The subjective probability of Steffi winning her match tomorrow is the ratio of favorable outcomes to the total number of outcomes. In this case, the probability is \(\frac{7}{12}\). The subjective probability that she will win her match tomorrow is \(\frac{7}{12}\).

Key Concepts

Odds in Favor

Odds in favor of an event provide a way to express the likelihood of that event happening compared to it not happening. Imagine a scale balancing two possibilities: 'success' on one side and 'failure' on the other. The 'odds in favor' tip this scale by showing how much weight is on the 'success' side.

Odds are different from probability but are related. They are usually expressed in the form of 'successes' to 'failures', or in our case, the likelihood of Steffi winning to her not winning. Here, the given odds are 7 to 5. To understand this better, picture seven instances where Steffi wins for every five instances where she doesn't. This ratio simplifies the understanding of the event's favorability without delving into the exact likelihood, which is what probability does.

Understanding odds is crucial because they provide a foundational grasp of how likely an event is to occur, allowing us to make informed guesses or decisions. When odds are high in favor, it signifies a higher chance of occurrence and vice versa.

Total Number of Outcomes

The concept of 'total number of outcomes' is a key element in calculating probability. It's the foundation of what makes probability work, as it gives us the complete set of possible scenarios we can expect from an event.

In probability, every possible outcome must be accounted for to gauge an accurate likelihood of any particular outcome occurring. For Steffi’s match, the total number of outcomes is determined by adding the favorable outcomes (7 wins) to the unfavorable outcomes (5 losses), which equals 12 possible outcomes. This addition ensures that all conceivable results are included in our calculation, providing a full view of the event's landscape.

Whether tossing a coin, rolling a die, or predicting the results of a tennis match, understanding the total number of outcomes grounds us in the realm of what's possible, offering a clear roadmap to navigate probability's often intricate pathways.

Ratios in Probability

Ratios in probability are used to compare the number of ways an event can occur to the total number of outcomes. They are essentially the building blocks we use to quantify probability, turning qualitative observations into quantitative facts.

For instance, Steffi’s probability of winning, as highlighted in the exercise solution, is calculated by dividing the number of favorable outcomes (7 wins) by the total number of outcomes (12). This results in a ratio, or fraction, of \(\frac{7}{12}\), which tells us out of the full range of possible outcomes, how many of those are wins for Steffi.

Using ratios empowers us with precision. It takes the broad concept of 'chance' and breaks it down into exact figures, allowing for a more nuanced understanding. As students work with these ratios, they begin to see probability not as an abstract concept but as a tangible and calculable aspect of everyday life.