Problem 99
Question
The figure shows that when a die is rolled, there are six equally likely outcomes: 1, 2, 3, 4, 5, or 6. What fraction of the outcomes is less than 5?
Step-by-Step Solution
Verified Answer
The fraction of outcomes that are less than 5 is \(\frac{4}{6}\), which simplifies to \(\frac{2}{3}\).
1Step 1: Understand the Outcomes of the Die Roll
Recognize that when a six-sided die is rolled, there are six equally likely outcomes: 1, 2, 3, 4, 5 or 6.
2Step 2: Determine the Desired Outcomes
Identify the outcomes that fulfill our condition, which is being less than 5. These are 1, 2, 3, and 4. Thus, we have four desired outcomes.
3Step 3: Compute the Probability
Probability is calculated as the number of desired outcomes divided by the total number of outcomes. So divide the number of desired outcomes (4) by the total outcomes (6) to get the fraction.
Key Concepts
Outcomes of a Die RollCalculating ProbabilityFraction of Outcomes
Outcomes of a Die Roll
When you roll a standard six-sided die, there are six possible outcomes, and these outcomes are the numbers 1 through 6. One of the fundamental principles of probability is the idea of equally likely outcomes. In the case of a fair die, this means each number has an equal chance of landing face up, making the probability of any specific result 1 in 6.
To get a firm grasp on this concept, picture a die in your hand. Each face is different, and when you roll it, the die has to settle on one of those six faces, giving you those six different possible results. These outcomes form the basis for calculating the probabilities of various events related to dice rolls, such as getting an even number, a number greater than 3, or in this case, a number less than 5.
Understanding the specific outcomes, and knowing that each one is just as likely as the others, is key to delving deeper into the world of probability.
To get a firm grasp on this concept, picture a die in your hand. Each face is different, and when you roll it, the die has to settle on one of those six faces, giving you those six different possible results. These outcomes form the basis for calculating the probabilities of various events related to dice rolls, such as getting an even number, a number greater than 3, or in this case, a number less than 5.
Understanding the specific outcomes, and knowing that each one is just as likely as the others, is key to delving deeper into the world of probability.
Calculating Probability
The core of probability calculation is surprisingly straightforward. You simply take the number of desired outcomes and divide it by the total number of possible outcomes. For a single die roll, considering all outcomes are equally likely, the probability of an event is given by the formula:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]
In our example where we want to find the probability of rolling a number less than 5, we identify that 1, 2, 3, and 4 are the only numbers fitting that description. With this clear understanding, we proceed to the calculation step, which is incredibly straightforward once you know what numbers to plug into the equation. This method can be applied to any scenario where you need to find the probability of a certain event from a set of distinct, equally likely possibilities.
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]
In our example where we want to find the probability of rolling a number less than 5, we identify that 1, 2, 3, and 4 are the only numbers fitting that description. With this clear understanding, we proceed to the calculation step, which is incredibly straightforward once you know what numbers to plug into the equation. This method can be applied to any scenario where you need to find the probability of a certain event from a set of distinct, equally likely possibilities.
Fraction of Outcomes
A fraction represents a part of a whole. In probability, the fraction of outcomes refers to how many parts of the whole fit the criteria we are interested in. For instance, if we want to determine the fraction of outcomes where a die roll results in a number less than 5, we consider each outcome that satisfies the condition as a part of the fraction.
For our die roll, since 1, 2, 3, and 4 are the outcomes that are less than 5, we have four parts out of the six possible outcomes. This fraction is written as \( \frac{4}{6} \). However, it is often helpful to simplify fractions to their lowest terms to make the information more digestible and easier to compare to other fractions. In this case, \( \frac{4}{6} \) simplifies to \( \frac{2}{3} \), as both the numerator (4) and the denominator (6) can be divided by 2.
Picturing the die roll outcomes as slices of a pie can help you visualize the fraction of outcomes. If the entire pie represents all possible results from rolling the die, then the part of the pie that fits our condition (less than 5) would be equivalent to the fraction \( \frac{4}{6} \), or more simply, \( \frac{2}{3} \) of the pie.
For our die roll, since 1, 2, 3, and 4 are the outcomes that are less than 5, we have four parts out of the six possible outcomes. This fraction is written as \( \frac{4}{6} \). However, it is often helpful to simplify fractions to their lowest terms to make the information more digestible and easier to compare to other fractions. In this case, \( \frac{4}{6} \) simplifies to \( \frac{2}{3} \), as both the numerator (4) and the denominator (6) can be divided by 2.
Picturing the die roll outcomes as slices of a pie can help you visualize the fraction of outcomes. If the entire pie represents all possible results from rolling the die, then the part of the pie that fits our condition (less than 5) would be equivalent to the fraction \( \frac{4}{6} \), or more simply, \( \frac{2}{3} \) of the pie.
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