4. Probability
Basic Concepts of Probability
1:36 minutesProblem 4.1.3c
Textbook Question
Dice and Coins
c. Find the probability that when a six-sided die is rolled, the outcome is 7.
Verified step by step guidance1
Understand the problem: A six-sided die has faces numbered 1 through 6. The problem asks for the probability of rolling a 7, which is outside the range of possible outcomes for this die.
Recall the definition of probability: Probability is calculated as the ratio of favorable outcomes to the total number of possible outcomes. Mathematically, \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \).
Identify the favorable outcomes: Since a six-sided die only has numbers 1 through 6, there are no favorable outcomes for rolling a 7. This means the number of favorable outcomes is 0.
Determine the total number of possible outcomes: For a six-sided die, the total number of possible outcomes is 6 (one for each face).
Substitute the values into the probability formula: Using \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \), substitute 0 for the numerator and 6 for the denominator to calculate the probability.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability
Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. An event with a probability of 0 means it cannot happen, while a probability of 1 means it is certain to happen. In the context of rolling a die, the probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
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Sample Space
The sample space is the set of all possible outcomes of a random experiment. For a six-sided die, the sample space consists of the numbers 1 through 6, representing each face of the die. Understanding the sample space is crucial for calculating probabilities, as it provides the total number of outcomes against which favorable outcomes are compared.
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Favorable Outcomes
Favorable outcomes are the specific results of an experiment that satisfy the conditions of the event we are interested in. In the case of rolling a six-sided die, there are no favorable outcomes for rolling a 7, as the die only has faces numbered 1 to 6. This concept is essential for determining the probability of an event, as it directly influences the calculation of the probability ratio.
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