Problem 30 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The probability of an event \(E\) occurring is \(.8\). What are the odds in favor of \(E\) occurring? What are the odds against \(E\) occurring?

Step-by-Step Solution

Answer

The odds in favor of E occurring are 4:1 and the odds against E occurring are 1:4.

Step 1: Find the probability of E not occurring

Given, the probability of E occurring is 0.8. We know that the sum of the probabilities of an event occurring and not occurring is always 1. So, the probability of E not occurring can be found as: \(P(E') = 1 - P(E)\) Substituting the value: \(P(E') = 1 - 0.8 = 0.2\)

Step 2: Calculate the odds in favor of E occurring

The odds in favor is the ratio of the probability of E occurring to the probability of E not occurring. We can write it as a fraction: Odds in favor of E = \(\frac{P(E)}{P(E')}\) Substitute the values of the probabilities: Odds in favor of E = \(\frac{0.8}{0.2}\) Simplify the fraction: Odds in favor of E = \(\frac{4}{1}\) or 4:1

Step 3: Calculate the odds against E occurring

The odds against is the reciprocal of the odds in favor, that is, the ratio of the probability of E not occurring to the probability of E occurring. We can write it as a fraction: Odds against E = \(\frac{P(E')}{P(E)}\) Substitute the values of the probabilities: Odds against E = \(\frac{0.2}{0.8}\) Simplify the fraction: Odds against E = \(\frac{1}{4}\) or 1:4

Step 4: Final Answer

The odds in favor of E occurring are 4:1 and the odds against E occurring are 1:4.

Key Concepts

Odds

Odds are a way to express the likelihood of an event occurring compared to it not occurring. Unlike probability, which is about the chance of a single event, odds involve two opposing outcomes. To find the odds in favor of an event, you compare the chances of it happening to the chances of it not happening.

Here's how you do it:

• Odds in Favor = Probability of Event Occurring / Probability of Event Not Occurring

Odds can be expressed in a ratio format, such as 4:1. This means, for every 4 times the event occurs, it fails to occur once.

There are also odds against, which are the reverse:

• Odds Against = Probability of Event Not Occurring / Probability of Event Occurring

In our example with event E, the odds in favor are 4:1, and the odds against are 1:4.
Understanding odds is especially useful in games, sports, and risk assessments where outcomes are uncertain.

Event Occurrence

An event occurrence refers to whether or not an event has happened. In probability terms, it stands for the outcome of a particular situation. Each event has a certain chance of taking place. If you have a probability, like in our example, where the event E has a probability of 0.8, it means:

• E occurs in 80% of scenarios, given the same conditions.
• The probability of E not occurring is simply the complementary probability, calculated as 1 minus the probability of occurrence.

So, if E occurs 80% of the time, it doesn't occur 20% of the time.

This fundamental understanding helps in calculating probabilities and understanding real-world scenarios, like weather predictions, game outcomes, and other uncertain events.

Ratio

Ratios are a way to show the relationship between two numbers, expressing how many times one value contains or is contained within the other. In probability, ratios are used to compare two different probabilities.

For example, in our event E, the probability of 0.8 and 0.2 can be expressed as a ratio, showing a relationship between outcomes:

• The ratio of probabilities (0.8:0.2) simplifies to 4:1, which means for every 4 successes, there is 1 failure.

Ratios provide a clear, concise way to understand and communicate the likelihood of events. They can be crucial in making predictions, assessing risks, and deciding actions in various fields from finance to science.