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Chapter 12: Problem 10

Compute the probability of tossing a six-sided die and getting a number less than \(3 .\)

Short Answer

Expert verified
The probability of tossing a number less than 3 is \( \frac{1}{3} \).

Step by step solution

01

Identify the Total Outcomes

When tossing a six-sided die, there are 6 possible outcomes one can get, corresponding to each face of the die: 1, 2, 3, 4, 5, and 6.
02

Determine the Favorable Outcomes

For this problem, we are interested in the outcomes that are less than 3. These outcomes are 1 and 2. So, there are 2 favorable outcomes.
03

Calculate the Probability

The probability of an event is calculated using the formula \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \). Here, it will be \( P(<3) = \frac{2}{6} = \frac{1}{3} \).

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Key Concepts

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

six-sided die
When discussing a six-sided die, we're referring to one of the most common tools used in probability exercises. A die is a cube, often used in board games, with each face numbered from 1 to 6.
  • Each face represents a unique number.
  • The numbers on a six-sided die are 1, 2, 3, 4, 5, and 6.
  • The die is fair, meaning each face has an equal chance of landing face up.
It's crucial to understand these basics because the number of faces on the die represents the total "possible outcomes" when you roll it.
favorable outcomes
In probability, "favorable outcomes" refer to the specific results that satisfy the condition described in the problem. For instance, when asked to find the probability of rolling a number less than 3 on a six-sided die, the favorable outcomes would be those outcomes less than 3.
  • In this scenario, the favorable outcomes are 1 and 2.
  • Count these outcomes to determine the number of favorable cases.
Here, since both 1 and 2 fulfill the condition, we have 2 favorable outcomes.
possible outcomes
The term "possible outcomes" comes from the total number of results you can get when an action is performed. For a six-sided die, each roll results in one of six distinct outcomes corresponding to its faces.
  • Possible outcomes for a die: 1, 2, 3, 4, 5, and 6.
  • This total number (six in this case) is used as the denominator in the probability formula.
Understanding "possible outcomes" helps us set the stage for calculating probabilities by comparing it to the favorable outcomes.
probability formula
The probability formula is a fundamental concept used to determine the likelihood of an event occurring. It is expressed as:\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]Here's a quick breakdown of how it's applied:
  • Identify the total number of outcomes. For a six-sided die, this is 6.
  • Count the favorable outcomes. In our example, numbers less than 3 are 1 and 2, so there are 2 favorable outcomes.
  • Plug these numbers into the formula to find probabilities. Here, the probability of getting a number less than 3 is \( \frac{2}{6} = \frac{1}{3} \).
This formula provides a straightforward method to quantify uncertainty and predict the likelihood of specific outcomes in probability exercises.

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