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Chapter 9: Problem 18

A standard number cube is tossed. Find each probability. \(P(3 \text { or odd })\)

Short Answer

Expert verified
The probability of rolling a '3' or 'an odd number' when tossing a standard die is 0.5.

Step by step solution

01

Determine the total number of outcomes

When a die is tossed, there are six possible outcomes. These outcomes are the numbers 1 through 6.
02

Determine the number of favorable outcomes

The favorable outcomes for this particular exercise are those that result in the roll of a '3' or 'an odd number'. The odd numbers on a die are 1, 3, and 5. Therefore, there are three favorable outcomes鈥�1, 3, and 5.
03

Determine the probability

The probability of an event is determined by dividing the number of favorable outcomes by the total number of outcomes. In this case, the number of favorable outcomes is 3 (i.e., rolling a 1, 3 or 5), and the total number of outcomes is 6. Thus, the probability (\(P\)) is calculated as \(P = \frac{favorable \ outcomes}{total \ outcomes}= \frac{3}{6} = 0.5\)

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Key Concepts

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

Number Cube
A number cube, commonly known as a die, is a small cube with numbers written on its faces. Each face of a standard number cube shows one of the numbers from 1 to 6. When you toss a number cube, any of these six numbers may appear on the top face, making this a classic example of a simple random experiment in probability. The number cube is an essential tool in teaching the foundational concepts of probability because it provides a controlled set of potential outcomes that are easy to observe and calculate.
Favorable Outcomes
In probability, favorable outcomes are those specific results that satisfy the conditions of a given event. When considering the probability of rolling a '3' or an 'odd number' on a number cube, we carefully determine which outcomes meet these criteria. On a standard number cube:
  • A '3' is clearly one of the numbers.
  • The odd numbers are 1, 3, and 5.
Since '3' is both a specific and an odd number, the favorable outcomes in this scenario are: rolling a 1, 3, or 5. With three possible favorable outcomes, we can then proceed to calculate the probability.
Total Number of Outcomes
To calculate the probability of an event, understanding the total number of possible outcomes is crucial. For a number cube, each face represents a potential outcome. Thus, the total number of outcomes when you toss a number cube is the total number of its faces: six. These outcomes are equally likely, represented by numbers 1 through 6. Knowing this, we can calculate probabilities with confidence since each outcome has an equal chance of occurring.
Odd Numbers
The concept of odd numbers is fundamental in probability exercises involving a number cube. Odd numbers are integers not evenly divisible by 2, meaning they leave a remainder of 1 when divided by 2. On a number cube marked with numbers 1 to 6, the odd numbers are 1, 3, and 5. Recognizing these numbers is key, especially when calculating probabilities involving specific conditions. When asked to find the probability of rolling an odd number, you count these odd-numbered faces as favorable outcomes against the total possible outcomes.

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