If events $$ A $$ and $$ B $$ are from the same sample space, and if $$ P(A)=0.8 $$ and $$ P(B)=0.9 $$, can events $$ A $$ and $$ B $$ be mutually exclusive? Explain. Choose the correct answer below. A. All probabilities are less than or equal to 1 and greater than or equal to 0 . If events are mutually exclusive, then $$ P(A \cup B)=P(A)+P(B) $$. Therefore, if $$ A $$ and $$ B $$ are mutually exclusive, $$ P(A \cup $$ $$ \begin{array}{l}B)=P(A)+P(B)=0.8+0.9=1 ext {, which is possible. Therefore, events } A ext { and } B ext { are mutually exclusive. } \ ext { B. All probabilities are less than or equal to } 1 ext { and greater than or equal to } 0 ext {. If events are mutually exclusive, } \ ext { then } P(A \cup B)=P(A)+P(B) ext {. Therefore, if } A ext { and } B ext { are mutually exclusive, } P(A \cup\end{array} $$ $$ B)=P(A)+P(B)=0.8+0.9=0 $$, which is possible. Therefore, events $$ A $$ and $$ B $$ are mutually exclusive. C. All probabilities are less than or equal to 1 and greater than or equal to 0 . If events are mutually exclusive, then $$ P(A \cup B)=P(A)+P(B) $$. Therefore, if $$ A $$ and $$ B $$ are mutually exclusive, $$ P(A \cup $$ B $$ \begin{array}{l}B)=P(A)+P(B)=0.8+0.9=1.7 ext {, which is impossible. Therefore, events } A ext { and } B ext { are not mutually exclusive. } \ ext { D. All probabilities are less than or equal to } 1 ext { and greater than or equal to } 0 ext {. If events are mutually exclusive, } \ ext { then } P(A \cup B)=P(A)+P(B) ext {. Therefore, if } A ext { and } B ext { are mutually exclusive, } P(A \cup \ B)=P(A)+P(B)=0.8+0.9=1.7 ext {, which is possible. Therefore, events } A ext { and } B ext { are mutually exclusive. }\end{array} $$

Question

If events $$ A $$ and $$ B $$ are from the same sample space, and if $$ P(A)=0.8 $$ and $$ P(B)=0.9 $$, can events $$ A $$ and $$ B $$ be mutually exclusive? Explain. Choose the correct answer below. A. All probabilities are less than or equal to 1 and greater than or equal to 0 . If events are mutually exclusive, then $$ P(A \cup B)=P(A)+P(B) $$. Therefore, if $$ A $$ and $$ B $$ are mutually exclusive, $$ P(A \cup $$ $$ \begin{array}{l}B)=P(A)+P(B)=0.8+0.9=1 	ext {, which is possible. Therefore, events } A 	ext { and } B 	ext { are mutually exclusive. } \ 	ext { B. All probabilities are less than or equal to } 1 	ext { and greater than or equal to } 0 	ext {. If events are mutually exclusive, } \ 	ext { then } P(A \cup B)=P(A)+P(B) 	ext {. Therefore, if } A 	ext { and } B 	ext { are mutually exclusive, } P(A \cup\end{array} $$ $$ B)=P(A)+P(B)=0.8+0.9=0 $$, which is possible. Therefore, events $$ A $$ and $$ B $$ are mutually exclusive. C. All probabilities are less than or equal to 1 and greater than or equal to 0 . If events are mutually exclusive, then $$ P(A \cup B)=P(A)+P(B) $$. Therefore, if $$ A $$ and $$ B $$ are mutually exclusive, $$ P(A \cup $$ B $$ \begin{array}{l}B)=P(A)+P(B)=0.8+0.9=1.7 	ext {, which is impossible. Therefore, events } A 	ext { and } B 	ext { are not mutually exclusive. } \ 	ext { D. All probabilities are less than or equal to } 1 	ext { and greater than or equal to } 0 	ext {. If events are mutually exclusive, } \ 	ext { then } P(A \cup B)=P(A)+P(B) 	ext {. Therefore, if } A 	ext { and } B 	ext { are mutually exclusive, } P(A \cup \ B)=P(A)+P(B)=0.8+0.9=1.7 	ext {, which is possible. Therefore, events } A 	ext { and } B 	ext { are mutually exclusive. }\end{array} $$

UpStudy AI · Accepted Answer

1. Since $$ P(A)=0.8 $$ and $$ P(B)=0.9 $$, if $$ A $$ and $$ B $$ are mutually exclusive, then they cannot occur simultaneously; thus, $$ P(A \cap B)=0 $$.

2. For mutually exclusive events, the probability of their union is given by
   $$
   P(A \cup B)=P(A)+P(B).
   $$

3. Substituting the given probabilities:
   $$
   P(A \cup B)=0.8+0.9=1.7.
   $$

4. However, a probability cannot exceed $$ 1 $$. Since $$ 1.7>1 $$, this situation is impossible.

5. Therefore, events $$ A $$ and $$ B $$ cannot be mutually exclusive.

The correct answer is **C**.
Events $$ A $$ and $$ B $$ cannot be mutually exclusive because the sum of their probabilities exceeds 1, which is impossible. Therefore, the correct answer is **C**.

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