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Chapter 12: Problem 18

What is the probability that a calendar is randomly turned to the month of January or April?

Short Answer

Expert verified
The probability of turning to January or April is \( \frac{1}{6} \).

Step by step solution

01

Determine Total Months

First, recognize that there are 12 months in a year. This will be the total number of outcomes when randomly opening a calendar.
02

Identify Favorable Outcomes

Identify the months we are interested in: January and April. There are 2 favorable outcomes: either landing on January or April.
03

Calculate Probability

Use the probability formula: \( P = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \). Here, \( P = \frac{2}{12} \).
04

Simplify the Fraction

Simplify \( \frac{2}{12} = \frac{1}{6} \) by dividing both the numerator and denominator by their greatest common divisor, which is 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Favorable Outcomes
When calculating probability, one of the first steps is to determine the favorable outcomes. These are the specific results that fit the criteria of what you are trying to calculate. In our example of the calendar being randomly turned, the favorable outcomes are the months of January or April. This means anytime the calendar is opened to either of these months, it counts as a favorable outcome.
  • Favorable outcomes are always a subset of the total possible outcomes.
  • Identifying favorable outcomes accurately is essential for solving any probability problem correctly.
In the context of our exercise, there are 2 favorable outcomes: January and April.
Introducing the Probability Formula
Probability helps us understand how likely an event is to occur. To compute it, we use the probability formula which is:
  • P = \( \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Outcomes}} \)
This formula compares the number of ways an event can happen (favorable outcomes) to the total number of possibilities (total outcomes).
In our calendar example, there are 12 total months or total outcomes. As we determined earlier, there are 2 favorable outcomes. Plugging these numbers into the formula gives:
  • \( P = \frac{2}{12} \)
This step-by-step approach allows us to methodically tackle probability problems by focusing on relevant parts.
Simplifying Fractions
Once we have the fraction from the probability formula, simplifying it makes it easier to understand. To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD). This process reduces the fraction to its simplest form.

Looking at our example, the fraction \( \frac{2}{12} \) can be simplified. The greatest common divisor of 2 and 12 is 2. So, we divide both by 2:
  • Numerator: \( \frac{2}{2} = 1 \)
  • Denominator: \( \frac{12}{2} = 6 \)
Thus, \( \frac{2}{12} \) simplifies to \( \frac{1}{6} \). Simplifying fractions ensures results are in their most straightforward form, making them easier to interpret in the context of probability.

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