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 for three consecutive years?

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Step 1: Understand the problem. The probability of a flood in any given year is 0.2, and we are tasked with finding the probability of a flood occurring in three consecutive years. This involves multiplying probabilities because the events are independent.

Step 2: Recall the rule for independent events. If events are independent, the probability of all events occurring is the product of their individual probabilities. Mathematically, this is expressed as P(A and B and C) = P(A) × P(B) × P(C).

Step 3: Assign the probability for each year. Since the probability of a flood in any given year is 0.2, we can write P(flood in year 1) = 0.2, P(flood in year 2) = 0.2, and P(flood in year 3) = 0.2.

Step 4: Multiply the probabilities. Using the formula for independent events, calculate the probability of a flood in three consecutive years as P(flood in year 1) × P(flood in year 2) × P(flood in year 3).

Step 5: Write the final expression. The probability of a flood for three consecutive years is expressed as 0.2 × 0.2 × 0.2. Simplify this expression to find the result.

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Probability Basics

Probability is a measure of the likelihood that an event will occur, expressed as a number between 0 and 1. A probability of 0 indicates an impossible event, while a probability of 1 indicates a certain event. In this context, the probability of a flood occurring in any given year is 0.2, meaning there is a 20% chance of a flood happening.

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Introduction to Probability

Independent Events

Independent events are those whose outcomes do not affect each other. In this scenario, the occurrence of a flood in one year does not influence the probability of a flood in the following years. Therefore, to find the probability of floods occurring over multiple years, we can multiply the probabilities of each individual event.

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Probability of Multiple Independent Events

Multiplication Rule of Probability

The multiplication rule states that the probability of two or more independent events occurring together is the product of their individual probabilities. For three consecutive years with a flood probability of 0.2, the overall probability of floods in all three years is calculated as 0.2 multiplied by itself three times, or (0.2)^3.

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Probability of Multiple Independent Events

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Domanda del libro di testo

Retaining the Concepts. Expand: $\log_{7} (\frac{5 \sqrt{x}}{49 y^{10}})$