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Chapter 9: Problem 23

If a die is tossed, what is the probability that a number from 1 to 6 will come up?

Short Answer

Expert verified
The probability is 1.

Step by step solution

01

Understanding the Problem

A die has 6 faces numbered from 1 to 6. We need to find the probability of getting any one of these numbers when the die is tossed.
02

Defining Total Outcomes

Identify the total number of possible outcomes when a die is tossed. Since a die has 6 faces, the total number of outcomes is 6.
03

Defining Favorable Outcomes

Identify the number of favorable outcomes. Since we want any number from 1 to 6, every face of the die is a favorable outcome. So, there are 6 favorable outcomes.
04

Calculating the Probability

The probability of an event is defined as the number of favorable outcomes divided by the total number of possible outcomes. Therefore, the probability is \(\frac{6}{6} = 1\).

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Key Concepts

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

Understanding Basic Probability
Probability measures how likely an event is to happen. It ranges from 0 to 1. A probability of 0 means the event cannot happen, while a probability of 1 means it will definitely happen. We commonly express probabilities as fractions, decimals, or percentages. For example, flipping a coin and getting heads has a probability of \( \frac{1}{2} \) or 0.5 or 50%.
Probability helps us predict outcomes in various everyday situations, from games to weather forecasts. When dealing with basic probability problems, it鈥檚 important to clearly identify both the total number of outcomes and the favorable ones.
Defining Favorable Outcomes
Favorable outcomes are the specific outcomes we are interested in within an experiment or situation. Let鈥檚 imagine rolling a fair 6-sided die, like in our exercise. Each number from 1 to 6 on the die is an outcome.
If we want to roll an even number, the favorable outcomes are 2, 4, and 6 - three outcomes in total. Similarly, if we want any number from 1 to 6 (like in the exercise), all six faces are favorable outcomes. The better we define what counts as a favorable outcome, the more accurate our probability calculations will be.
Here's a tip: Always ensure that each outcome you list actually aligns with what you鈥檙e trying to find the probability for!
Calculating Total Outcomes
Total outcomes refer to all possible outcomes in an experiment. This number is crucial in probability calculations because it forms the denominator in our probability fraction.
In our die example, there are 6 possible outcomes because a die has 6 faces.
When calculating the probability, if we have six favorable outcomes (getting any number from 1 to 6), and the total outcomes are also six, the probability is \( \frac{6}{6} = 1 \). Remember:
  • Total outcomes should account for all possibilities, regardless of whether they are favorable or not.
  • In different scenarios like drawing cards from a deck or flipping two coins, the total outcomes might differ, so always double-check your total count based on the experiment setup.

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