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If A and $\mathrm{B}$ are mutually exclusive events, $\mathrm{P}(\mathrm{A})=0.35$ and $P(B)=0.45$, find
(a) $\mathrm{P}\left(\mathrm{A}^{\prime}\right)$
(b) $\quad \mathrm{P}\left(\mathrm{B}^{\prime}\right)$
(c) $\mathrm{P}(\mathrm{A} \cup \mathrm{B})$
(d) $\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
(e) $\quad P\left(A \cap B^{\prime}\right)$
(f) $\mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}\right)$
(a) $\mathrm{P}\left(\mathrm{A}^{\prime}\right)$
(b) $\quad \mathrm{P}\left(\mathrm{B}^{\prime}\right)$
(c) $\mathrm{P}(\mathrm{A} \cup \mathrm{B})$
(d) $\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
(e) $\quad P\left(A \cap B^{\prime}\right)$
(f) $\mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}\right)$
Solution:
1758 Upvotes
Verified Answer
Give $P(A)=0.35 \& P(B)=0.45$
(a) $\mathrm{P}\left(\mathrm{A}^{\prime}\right)=1-\mathrm{P}(\mathrm{A})=1-0.35=0.65$
(b) $\mathrm{P}\left(\mathrm{B}^{\prime}\right)=1-\mathrm{P}(\mathrm{B})=1-0.45=0.55$
(c) $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$ $=0.35+0.45-0=0.8$
(d) $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0 \quad(\therefore$ Events are Mutually Exclusive $)$
(e) $\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}^{\prime}\right)=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.35-0=0.35$
(f) $\mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}\right)=\mathrm{P}(\mathrm{A} \cup \mathrm{B})^{\prime}=1-\mathrm{P}(\mathrm{A} \cup \mathrm{B})=1-0.8=0.2$
(a) $\mathrm{P}\left(\mathrm{A}^{\prime}\right)=1-\mathrm{P}(\mathrm{A})=1-0.35=0.65$
(b) $\mathrm{P}\left(\mathrm{B}^{\prime}\right)=1-\mathrm{P}(\mathrm{B})=1-0.45=0.55$
(c) $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$ $=0.35+0.45-0=0.8$
(d) $\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0 \quad(\therefore$ Events are Mutually Exclusive $)$
(e) $\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}^{\prime}\right)=\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.35-0=0.35$
(f) $\mathrm{P}\left(\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}\right)=\mathrm{P}(\mathrm{A} \cup \mathrm{B})^{\prime}=1-\mathrm{P}(\mathrm{A} \cup \mathrm{B})=1-0.8=0.2$
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