Textbook Question
The probability of a flood in any given year in a region prone to floods is 0.2. What is the probability of a flood two years in a row?
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10. Combinatorics & Probability
Probability
1:51 minutesProblem 89
Textbook Question
A die is rolled. Find the probability of getting a number less than 5.
Verified step by step guidance1
Step 1: Understand the problem. A standard die has 6 faces numbered 1 through 6. The goal is to find the probability of rolling a number less than 5.
Step 2: Identify the favorable outcomes. Numbers less than 5 on a die are 1, 2, 3, and 4. This gives us 4 favorable outcomes.
Step 3: Determine the total possible outcomes. Since a die has 6 faces, there are 6 possible outcomes in total.
Step 4: Use the probability formula: \( P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \). Substitute the values: \( P(E) = \frac{4}{6} \).
Step 5: Simplify the fraction if necessary. \( \frac{4}{6} \) can be simplified to \( \frac{2}{3} \). This represents the probability of rolling a number less than 5.
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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 will not happen, while a probability of 1 means it will certainly happen. In this context, we are interested in the probability of rolling a number less than 5 on a standard six-sided die.
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Sample Space
The sample space is the set of all possible outcomes of a random experiment. For a single roll of a six-sided die, the sample space consists of the numbers {1, 2, 3, 4, 5, 6}. Understanding the sample space is crucial for calculating probabilities, as it provides the total number of outcomes against which specific events can be measured.
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Favorable Outcomes
Favorable outcomes refer to the specific outcomes in the sample space that satisfy the condition of the event we are interested in. In this case, the favorable outcomes for rolling a number less than 5 are {1, 2, 3, 4}, which totals four outcomes. The probability of the event is calculated by dividing the number of favorable outcomes by the total number of outcomes in the sample space.
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