Problem 32 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The odds in favor of an event \(E\) occurring are 9 to 7 . What is the probability of \(E\) occurring?

Step-by-Step Solution

Answer

The probability of \(E\) occurring is \(\frac{9}{16}\).

Step 1: Identify the favorable and unfavorable outcomes

The favorable outcomes are the chances of the event \(E\) occurring (9) and the unfavorable outcomes are the chances of the event \(E\) not occurring (7).

Step 2: Calculate the total possible outcomes

The total possible outcomes are the sum of the favorable and unfavorable outcomes. In this case, the total possible outcomes are 9 + 7 = 16.

Step 3: Calculate the probability of \(E\) occurring

The probability of event \(E\) occurring can be found by dividing the number of favorable outcomes by the total number of possible outcomes. Therefore, the probability of \(E\) occurring is: \( P(E) = \frac{\text{favorable outcomes}}{\text{total possible outcomes}} = \frac{9}{16} \) So, the probability of event \(E\) occurring is \(\frac{9}{16}\).

Key Concepts

Odds

Odds are a way of expressing the likelihood that a particular event will happen compared to it not happening. The term "odds" specifically refers to the ratio of favorable outcomes to unfavorable outcomes. For example, if the odds in favor of an event are 9 to 7, this means there are 9 favorable outcomes for every 7 unfavorable outcomes.

Understanding odds helps in making sense of how likely an event is to occur, offering another perspective apart from probability. Here’s how it works:

• Identify the outcome: Determine what is the favorable and what is the unfavorable outcome for the event in question.
• Calculate the odds: Express the likelihood as a ratio by comparing the favorable outcomes against the unfavorable ones.

Odds provide an intuitive view of chance, making them useful in areas like gambling and statistics where predictions are necessary.

Favorable Outcomes

Favorable outcomes are the scenarios in which the event of interest occurs. In probability, knowing the number of favorable outcomes helps in determining how likely the event is to happen.

To find favorable outcomes in any situation, ask yourself: "How many ways can this event happen?" Let's break it down:

• Define the event: Clearly understand the event whose occurrence you are evaluating.
• Count the favorable outcomes: List out all the scenarios that will lead to the event happening successfully.

In our example, 9 represents the favorable outcomes for event \(E\). Thus, these outcomes contribute to making the event happen, and they are crucial in the calculation of probability.

Unfavorable Outcomes

Unfavorable outcomes are those events where the desired outcome does not occur. In probability, these outcomes are just as important to consider as the favorable ones for accurate calculations.

Understanding unfavorable outcomes involves two key steps:

• Define non-occurrences: Identify what constitutes the event not happening.
• Count the outcomes: Determine how many ways the event can fail to occur.

In the exercise, the 7 represents the unfavorable outcomes for event \(E\). Knowing both favorable and unfavorable outcomes allows us to calculate the total possible outcomes, which is essential in finding the probability.