Problem 33 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The odds against an event \(E\) occurring are 2 to \(3 .\) What is the probability of \(E\) not occurring?

Step-by-Step Solution

Answer

The probability of event E not occurring is \(\frac{2}{5}\).

Step 1: Convert the odds to probability

To convert the odds against event E occurring, which are given as 2 to 3, to probability, we use the following formula: Probability of E = \(\frac{Odds\: in\: favor\: of\: E}{Odds\:in\:favor\:of\:E + Odds\:against\:E}\) In this case, the odds in favor of E are 3 (since the odds against E are 2 to 3), and the odds against E are 2.

Step 2: Calculate the probability of E

Now, we can plug the odds in favor and against E into the formula to calculate the probability of E occurring: Probability of E = \(\frac{3}{3+2}\) Probability of E = \(\frac{3}{5}\)

Step 3: Calculate the probability of E not occurring

The probability of E not occurring is the complement of the probability of E occurring. To find the complement of a probability, we can subtract the probability from 1: Probability of E not occurring = 1 - Probability of E Probability of E not occurring = 1 - \(\frac{3}{5}\)

Step 4: Simplify the result

Now we can simplify the result to get the probability of E not occurring: Probability of E not occurring = \(\frac{5}{5}\) - \(\frac{3}{5}\) Probability of E not occurring = \(\frac{2}{5}\) So, the probability of event E not occurring is \(\frac{2}{5}\).

Key Concepts

Odds to Probability Conversion

Understanding how to convert odds to probability is a fundamental skill in statistics and probability theory. When we talk about 'odds', we typically refer to odds in favor or against a particular event, like winning a game or drawing a red card from a deck. For example, if the odds against event E occurring are given as 2 to 3, it means that for every 2 instances the event does not happen, it happens 3 times.

To convert odds to probability, you can use the following formula:
\[\text{Probability of E} = \frac{\text{Odds in favor of E}}{\text{Odds in favor of E} + \text{Odds against E}}\]
Using this formula ensures that we have a clear understanding of the likelihood of an event occurring in terms of probability, which is always a number between 0 and 1, where 0 means the event will never happen, and 1 means it will happen for sure.

Complementary Probability

The concept of complementary probability is an incredibly useful tool in the world of probability calculations. The 'complement' of an event is simply whatever outcome does not constitute the event occurring. For any event E, the probability of E occurring plus the probability of E not occurring always equals 1. This is because the probabilities of all possible outcomes must add up to 1, the certainty that one outcome or the other will occur.

So, if you have the probability of event E occurring, calculating the probability of it not occurring is straightforward: simply subtract the probability of E from 1, like so:
\[\text{Probability of E not occurring} = 1 - \text{Probability of E}\]
Understanding complementary probability is crucial, especially when it's easier to calculate the chances of an event not happening than the event itself, or vice versa.

Probability Simplification

After you've converted odds to probability or calculated the complementary probability, your final step will often involve simplifying the probability to its lowest terms. This simplification process is an exercise in basic fraction reduction. The goal is to make the probability value as easy to understand and interpret as possible.

For example, with a probability expressed as \(\frac{3}{5}\), there's no need to simplify further since that's already in its simplest form. However, if we have the probability of an event not occurring as \(\frac{5}{5} - \frac{3}{5}\), we would simplify it to \(\frac{2}{5}\). Simplifying probabilities makes them easier to compare and work with, especially when you're dealing with multiple probabilities at once or when applying these probabilities to real-life scenarios.