Problem 29 - Algebra for College Students

Question

If the probability is 60% that the eye of Hurricane Edna comes ashore within 30 miles of Charleston, then what are the odds in favor of the eye of Edna coming ashore within 30 miles of Charleston?

Step-by-Step Solution

Answer

The odds in favor are 3:2.

Step 1: Understand Probability

Probability is given as a percentage. In this problem, the probability that the eye of Hurricane Edna comes ashore within 30 miles of Charleston is 60%, which can be written as a fraction: \[ P(A) = \frac{60}{100} = 0.6 \]

Step 2: Determine the Complementary Probability

Identify the probability that the event does not occur. If the probability of the event occurring is 0.6, then the probability of the event not occurring is \[ P(A^c) = 1 - P(A) = 1 - 0.6 = 0.4 \]

Step 3: Convert Probability to Odds

Odds in favor of an event are given as the ratio of the probability of the event occurring to the probability of it not occurring. Thus, the odds in favor of the eye of Hurricane Edna coming ashore within 30 miles of Charleston are \[ \text{Odds in favor} = \frac{P(A)}{P(A^c)} = \frac{0.6}{0.4} \]

Step 4: Simplify the Ratio

Simplify the ratio \( \frac{0.6}{0.4} \) to its lowest terms. \[ \frac{0.6}{0.4} = \frac{6}{4} = \frac{3}{2} \]. Hence, the odds in favor are 3:2.

Key Concepts

Probability

Probability helps us to measure the likelihood of an event. It tells us how often we can expect an event to happen in the long run. In the exercise, the probability of Hurricane Edna coming ashore within 30 miles of Charleston is given as 60%. We can express this as a fraction: \( \frac{60}{100} \), which simplifies to 0.6. This means that if we had 100 such hurricanes, we would expect 60 to come within 30 miles of Charleston.

Probability is always between 0 and 1, where 0 means the event is impossible and 1 means it's certain. A probability closer to 0 means the event is less likely, while a probability closer to 1 means it is more likely.

Complementary Probability

Complementary probability is about finding the chance of the opposite event happening. If something has a certain probability of happening, then there's always the opposite probability of it not happening. This is because all probabilities in a given situation must add up to 1.

In the problem, the probability of the hurricane coming ashore within 30 miles is 0.6. Therefore, the probability of it *not* coming ashore within 30 miles is: \( P(A^c) = 1 - 0.6 = 0.4 \)

Complementary probabilities are very useful because they often simplify the calculations, especially when working with odds.

Odds Ratio

The odds of an event represent the ratio of its probability to the probability of it not happening. It differs from probability but provides similar information in a different way.
In the exercise, the odds in favor of Hurricane Edna coming ashore within 30 miles of Charleston are calculated as: \( \text{Odds in favor} = \frac{P(A)}{P(A^c)} \). Knowing the probability of the event (0.6) and the complementary probability (0.4), the odds in favor are: \( \frac{0.6}{0.4} \).

Odds help us understand comparative likelihoods and are commonly used in areas like gambling and statistics.

Simplifying Ratios

To express odds in the simplest form, we often need to simplify the ratio. This step makes the numbers easier to interpret and compare.

In the problem, the odds ratio of 0.6 to 0.4 can be simplified. To simplify, we find the greatest common divisor (GCD) of the numerator and the denominator. Here, 0.6 and 0.4 can be converted to 6 and 4, respectively: \( \frac{6}{4} \).

We divide both the numerator and the denominator by their GCD, which is 2:
\( \frac{6 \div 2}{4 \div 2} = \frac{3}{2} \)
The simplified ratio is 3:2. Simplifying makes the ratio easier to grasp and more visually intuitive.