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Chapter 7: Problem 100

Explain how the property \(P\left(A^{\prime}\right)=1-P(A)\) follows directly from the properties of a probability distribution.

Short Answer

Expert verified
The property \(P(A') = 1 - P(A)\) follows directly from the properties of a probability distribution by using the basic properties of probabilities and the concept of complementary events. Since A and \(A'\) are mutually exclusive, their probabilities add up to the probability of the entire sample space, which is 1. Therefore, \(P(A) + P(A') = 1\), and rearranging this equation gives us \(P(A') = 1 - P(A)\).

Step by step solution

01

Property 1: Range of Probability Values

Probabilities must fall in the range between 0 and 1, that is \(0 \leq P(A) \leq 1\).
02

Property 2: Probability of Sample Space

The probability of the entire sample space, denoted by S, is 1. That is, \(P(S) = 1\).
03

Property 3: Probability of the Union of Mutually Exclusive Events

If two events, A and B, are mutually exclusive (i.e., they cannot both occur at the same time), then the probability of the union of A and B is the sum of their individual probabilities. That is, \(P(A \cup B) = P(A) + P(B)\), provided that \(A \cap B = \emptyset\). Now, let's use these properties to derive the desired property, \(P(A') = 1 - P(A)\).
04

Define the Complement of an Event

The complement of an event A, denoted as \(A'\), is the set of all outcomes in the sample space S that are not in A. In other words, A and \(A'\) are mutually exclusive and their union is the entire sample space, S. Formally, \(A \cap A' = \emptyset\) and \(A \cup A' = S\).
05

Apply Property 3

Since A and \(A'\) are mutually exclusive, we can apply Property 3, that states the probability of the union of mutually exclusive events is the sum of their probabilities: \(P(A \cup A') = P(A) + P(A')\).
06

Use Property 2

Recall that the union of A and \(A'\) is the entire sample space, S. Therefore, \(P(A \cup A') = P(S)\). Now, applying Property 2, we know that \(P(S) = 1\).
07

Find P(A')

We can now put together Steps 2 and 3 : \(P(A) + P(A') = P(S) = 1\). To find the probability of the complement of A, i.e., \(P(A')\), we can simply rearrange this equation: \(P(A') = 1 - P(A)\). Thus, we have shown that the property \(P(A') = 1 - P(A)\) follows directly from the properties of a probability distribution.

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