Compound Probability
Compound probability can be defined as the probability of the two events which are independent. It can be defined as the multiplication of the probability of two events that are not dependent.
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Probability theory is a branch of mathematics that deals with the subject of probability. Although there are many different concepts of probability, probability theory expresses the definition mathematically through a series of axioms. Usually, these axioms express probability in terms of a probability space, which assigns a measure with values ranging from 0 to 1 to a set of outcomes known as the sample space. An event is a subset of these outcomes that is described.
Conditional Probability
By definition, the term probability is expressed as a part of mathematics where the chance of an event that may either occur or not is evaluated and expressed in numerical terms. The range of the value within which probability can be expressed is between 0 and 1. The higher the chance of an event occurring, the closer is its value to be 1. If the probability of an event is 1, it means that the event will happen under all considered circumstances. Similarly, if the probability is exactly 0, then no matter the situation, the event will never occur.
#4 please
![### Educational Website Content
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#### Problem 4
**Objective:** Find the sum of the convergent series. Express the result as an exact value.
\[
4. \quad \sum_{n=1}^\infty \frac{3}{9n^2 + 21n + 10}
\]
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#### Problem 5
**Objective:** Determine the number of terms required to approximate the sum of the series with an error of less than 0.0001.
\[
5. \quad \sum_{n=0}^\infty \frac{(-1)^{n+1}}{3n^4 - 5}
\]
---
**Explanations:**
- **Convergent Series:** A series is convergent if the sum of its infinite terms approaches a specific finite number. For problem 4, solve the given series and express the sum as an exact value.
- **Approximation of Series:** In problem 5, determine the number of terms needed so that the difference between the actual sum and the estimated sum is less than the specified error margin, in this case, 0.0001.
Understanding and solving these types of problems are crucial in advanced calculus and mathematical analysis. The solutions often involve recognizing patterns, using known convergence tests, or applying error approximation techniques.
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