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Chapter 7: Problem 49

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If \(A\) is a subset of \(B\) and \(P(B)=0\), then \(P(A)=0\).

Short Answer

Expert verified
The statement is true. If \(A\) is a subset of \(B\) and \(P(B) = 0\), then \(P(A) = 0\) because the condition \(P(A) \le P(B)\) is satisfied, and probabilities cannot be negative.

Step by step solution

01

Recall the definition of a subset

A set \(A\) is a subset of a set \(B\) if every element of \(A\) is also an element of \(B\). In other words, \(A \subseteq B\) means that if \(x \in A\), then \(x \in B\).
02

Recall the properties of probabilities

Probabilities are values that range from 0 to 1, i.e., \(0 \le P(E) \le 1\) where E represents any event. Additionally, if an event is a subset of another event, the probability of the first event cannot be greater than the probability of the second event. In other words, if \(A \subseteq B\), then \(P(A) \le P(B)\).
03

Apply the properties of probabilities to the given statement

Now we can apply these properties to the given statement. If \(A\) is a subset of \(B\) and \(P(B) = 0\), then, since probabilities cannot be negative, the only possible value for \(P(A)\) that satisfies the condition \(P(A) \le P(B)\) is \(P(A) = 0\).
04

Conclusion

Therefore, the statement is true. If \(A\) is a subset of \(B\) and \(P(B) = 0\), then \(P(A) = 0\).

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