Page 445 #6b An experiment in which the three mutually exclusive events A, B, and C form a partition of the uniform sample space S is depicted in the diagram below. (Round your answers to three decimal places.
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.
Tree diagram
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.
Page 445 #6b
An experiment in which the three mutually exclusive
The diagram is divided as follows:
- A rectangular space labeled \( S \) contains three vertical sections labeled \( A, B, \) and \( C \), each containing numerical values.
- Inside the rectangle, there is an oval labeled \( D \) with its own numerical values.
The numerical values are:
- In section \( A \) (outside the oval \( D \)): 25
- In section \( B \) (overlapping with \( D \)): 20 inside \( D \) and 20 outside \( D \)
- In section \( C \) (overlapping with \( D \)): 15 inside \( D \) and 20 outside \( D \)
(b) **Find \( P(B \mid D^c) = \)**
**Explanation:**
- \( D^c \) refers to the complement of the event \( D \), which includes all outcomes not in \( D \).
- The values relevant to \( D^c \) are those outside the oval, specifically: 25 from \( A \), 20 from \( B \), and 20 from \( C \).
- To calculate \( P(B \mid D^c) \), use the formula:
\[
P(B \mid D^c) = \frac{P(B \cap D^c)}{P(D^c)}
\]
- Calculate \( P(B \cap D^c) \), which is the probability of event \( B \) occurring outside \( D \): 20.
- Calculate \( P(D^c) \), the total of all probabilities outside \( D \): 25 (from \( A \)) + 20 (from \( B \)) + 20 (from \( C \)) = 65.
Finally, substituting these into the formula:
\[
P(B \mid D^c) = \frac{20}{65}
\]
- Simplify and round the calculation to three decimal places.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3cadaab7-0242-405d-ab75-d1a15c225b30%2Fb44a4f03-8706-460a-ba80-468d001bccd5%2Fnqlj0p_processed.png&w=3840&q=75)
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