Problem 69 - Algebra for College Students

Question

Suppose that there is a predicted \(40 \%\) chance of freezing rain. State the prediction in terms of the odds against getting freezing rain. 3 to 2

Step-by-Step Solution

Answer

The odds against getting freezing rain are 3 to 2.

Step 1: Convert Probability to Fraction Form

The given probability of freezing rain is 40\%. To convert this percentage to a fraction, we divide by 100: \( \frac{40}{100} = \frac{2}{5} \). This fraction represents the chance of freezing rain occurring.

Step 2: Calculate Odds Against

The odds against an event are calculated by comparing the probability of the event not happening to the probability of it happening. The probability of no freezing rain is \( 1 - \frac{2}{5} = \frac{3}{5} \). Thus, the odds against freezing rain are \( \frac{3}{5} : \frac{2}{5} \).

Step 3: Simplify the Odds

To simplify the odds \( \frac{3}{5} : \frac{2}{5} \), divide both fractions by the common denominator: \( 3 : 2 \).Therefore, the simplified odds against freezing rain are 3 to 2.

Key Concepts

Understanding Percentages for Probability

Percentages are a way of expressing numbers as a fraction of 100. They're commonly used in probability to illustrate chances in a familiar form. When we say there's a 40% chance of freezing rain, it means that out of every 100 possible scenarios, the expected occurrence of freezing rain is 40 times. It's essential to realize that percentages make it easier to communicate complex probabilities as they are straightforward to understand.

• To convert a percentage to a decimal, divide by 100. For example, 40% becomes 0.4.
• To convert a percentage to a fraction, also divide by 100. So, 40% is equivalent to \( \frac{40}{100} \) which simplifies to \( \frac{2}{5} \).

Using percentages, we can easily compare, interpret, and analyze probabilities in practical terms.

Exploring Odds Against an Event

The concept of odds against refers to the likelihood that a particular event will not happen compared to the likelihood that it will happen. Odds are usually expressed in the form of a ratio. For instance, if there's a 40% chance of having freezing rain, the odds against this event are a way of stating how much less likely the freezing rain is.
In our example:

• The probability of the event (freezing rain) happening is \( \frac{2}{5} \).
• The probability of the event not happening (no freezing rain) is \( 1 - \frac{2}{5} = \frac{3}{5} \).

The odds against freezing rain are the ratio of these two probabilities: \( \frac{3}{5} : \frac{2}{5} \). These odds allow us to understand that the event is less likely to happen than not in a clear manner.

The Art of Fraction Simplification

Simplifying fractions is about finding an equivalent fraction that has the smallest possible integer numerator and denominator. This process helps in making ratios and odds more straightforward to interpret. When dealing with probabilities and odds, simplifying fractions makes them easier to understand and communicate.
To simplify the odds against freezing rain from \( \frac{3}{5} : \frac{2}{5} \):

• Recognize that both fractions have the same denominator (5).
• Eliminate the denominators by comparing the numerators directly, obtaining \( 3 : 2 \).

The simplified odds are thus expressed as 3 to 2, providing a clear and concise way to convey the comparative likelihood of the event.