Problem 31 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The probability of an event \(E\) not occurring is .6. What are the odds in favor of \(E\) occurring? What are the odds against \(E\) occurring?

Step-by-Step Solution

Answer

The probability of event \(E\) occurring is \(0.4\), while the probability of it not occurring is \(0.6\). Thus, the odds in favor of \(E\) occurring are \(\frac{2}{3}\) or \(2:3\), and the odds against \(E\) occurring are \(\frac{3}{2}\) or \(3:2\).

Step 1: Find the probability of E occurring

We are given the probability of E not occurring, which we can represent as P(not E) = 0.6. Since the sum of the probabilities of an event occurring and not occurring is always 1, we can find the probability of E occurring by subtracting the given probability from 1: P(E) = 1 - P(not E) = 1 - 0.6 = 0.4

Step 2: Determine the odds in favor of E occurring

Odds in favor of an event occurring are calculated as a ratio of the probability of the event happening to the probability of the event not happening. In this case, we can represent the odds in favor of E occurring as: Odds in favor of E = P(E) / P(not E) Now, we can substitute the probabilities that we calculated in Step 1 to get: Odds in favor of E = \( \frac{0.4}{0.6} = \frac{2}{3} \) So, the odds in favor of E occurring are 2:3.

Step 3: Determine the odds against E occurring

Odds against an event occurring are the inverse of the odds in favor of the event occurring. In other words, we simply switch the values in the ratio that we calculated in Step 2. So, the odds against E occurring can be represented as: Odds against E = P(not E) / P(E) Substitute the probabilities that we calculated in Step 1: Odds against E = \( \frac{0.6}{0.4} = \frac{3}{2} \) So, the odds against E occurring are 3:2.

Key Concepts

Understanding Probability of an Event

The probability of an event refers to how likely it is for that event to occur. It is a fundamental concept in statistics and mathematics that quantifies the uncertainty of an occurrence. When we measure probability, we're looking at a scale from 0 to 1, where 0 indicates impossibility and 1 signifies certainty. For example, if we flip a fair coin, the probability of getting 'heads' is 0.5, since there is an equal chance for 'heads' as there is for 'tails.'

In our exercise, we are presented with the probability of an event not happening, which is given as 0.6. To find the probability of the event occurring, we employ a simple rule that the total probability of all possible outcomes must equal 1. Therefore, by subtracting the probability of the event not occurring from 1, we swiftly determine that the probability of it occurring is 0.4. This aligns with everyday thinking; if it's not likely to rain today (60%), then it is likely to be dry (40%).

Calculating Odds

Odds are another way of expressing the likelihood of an event, but they differ from probability. Odds are usually expressed as a ratio. For instance, if you're gambling, odds provide a direct measure of your potential winnings. When the odds are in favor of an event, they show the ratio of occurrences to non-occurrences. Conversely, odds against reveal the ratio of non-occurrences to occurrences.

To calculate the odds in favor of an event, we divide the event's probability by its complement's probability (the probability of the event not occurring). In the exercise, the odds in favor of the event occurring are obtained by dividing 0.4 (the probability of occurrence) by 0.6 (the probability of non-occurrence), simplifying to the ratio 2:3. This means for every three times the event does not occur, it can be expected to happen twice.

Event's Complement Probability

The complement of an event in probability theory is just the opposite situation of the event. If we have an event 'E', its complement, denoted by 'not E', covers all possibilities that 'E' does not. The probabilities of an event and its complement always add up to 1 because together they represent all possible outcomes.

For example, if we are considering the event 'E' of it raining today, then the complement 'not E' would include all weather scenarios where it does not rain. If the probability that it rains today is 0.4, then the probability of it not raining (the complement) is 0.6. This concept helps us calculate unknown probabilities when we have the probability of the complement, as demonstrated in the given exercise. By understanding that the total probability (event plus its complement) must be 1, we can quickly find the missing piece of our probability puzzle.