Problem 95 - College Algebra

Question

Exercises \(95-97\) will help you prepare for the material covered in the next section. The figure shows that when a die is rolled, there are six equally likely outcomes: \(1,2,3,4,5,\) or \(6 .\) Use this information to solve each exercise. (image can't copy) What fraction of the outcomes is less than \(5 ?\)

Step-by-Step Solution

Answer

The fraction of the outcomes which result in a number less than 5 is \(\frac{2}{3}\).

Step 1: Identify the total outcomes

When a six-sided die is rolled, there can be six possible outcomes: 1, 2, 3, 4, 5, or 6. These constitute our total outcomes.

Step 2: Identify the outcomes of interest

From the six possible outcomes, those numbers which are less than 5 are of our interest. These numbers are 1, 2, 3, and 4. So, there are total of four outcomes that are less than 5.

Step 3: Write the fraction

Now that we have identified our outcomes of interest and total outcomes, we can express our result as a fraction. The number of our interested outcomes will be the numerator and total outcomes will be the denominator of the fraction.

Step 4: Calculate the fraction

The fraction will be \(\frac{4}{6} \). This fraction can be simplified by dividing both the numerator and denominator by the highest common factor, which is 2. After simplifying, the fraction becomes \(\frac{2}{3}\).

Key Concepts

Understanding Basic Probability

When we speak of basic probability, we are referring to the likelihood of a specific event or outcome occurring. Probability is expressed as a number between 0 and 1, with 0 indicating impossibility and 1 representing certainty. In the context of dice rolls, because the die is fair, each of the six faces has an equal chance of landing face up. To find the probability of an event, we start by counting how many ways the event can happen and divide it by the total number of possible outcomes.

In our dice example, if we want to know the probability of rolling a number less than 5, we first identify all the outcomes that match our event (in this case, 1, 2, 3, and 4), then we divide this number by the total number of possible outcomes when a single die is rolled, which is 6. This approach forms the fundamental principle of calculating basic probability.

Fractional Outcomes in Dice Rolls

Furthermore, probability results can often be expressed as fractions. A fractional outcome is simply one way to represent a part of a whole. In dice rolls, the 'whole' is the total number of potential outcomes (6 sides of the die), and 'part' of it is the event we're interested in (rolling a number less than 5).

We say that the fraction of outcomes that are less than 5 is the number of outcomes less than 5 (our event) over the number of total possible outcomes. This yields a fraction, which can be immediately used for further calculation or analysis. In our dice case, there are 4 outcomes less than 5 which gives us a fraction of \(\frac{4}{6}\) when we select the numerator as 4 and the denominator as 6, the total possible outcomes.

Simplifying Fractions in Probability

Reducing fractions is an essential skill in probability and mathematics in general. Simplifying or reducing a fraction means transforming it into its simplest form, where the numerator and denominator are as small as possible but still share the same ratio. To do this, we find the Greatest Common Divisor (GCD) of both numbers, which is the largest number that evenly divides both.

For the fraction \(\frac{4}{6}\), the GCD is 2. When we divide both the top and the bottom by 2, we simplify the fraction to \(\frac{2}{3}\). This doesn't change the value of the fraction, just its appearance. Simplifying fractions can make them easier to understand and work with, especially when comparing probabilities or combining multiple probabilities.