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An examination consists of multiple-choice questions, each having five possible answers. Suppose you are a student taking the exam. and that you reckon you have probability $0.75$ of knowing the answer to any question that may be asked and that, if you do not know, you intend to guess an answer with probability $1 / 5$ of being correct. What is the probability you will give the correct answer to a question?
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Let $A$ be the event that you give the correct answer. Let $B$ be the event that you knew the answer. We want to find $\mathrm{P}(A) .$ But $\mathrm{P}(A)=\mathrm{P}(A \cap B)+\mathrm{P}\left(A \cap B^{c}\right)$ where $\mathrm{P}(A \cap B)=\mathrm{P}(A \mid B) \mathrm{P}(B)=1 \times 0.75=0.75$
and $\mathrm{P}\left(A \cap B^{c}\right)=\mathrm{P}\left(A \mid B^{c}\right) \mathrm{P}\left(B^{c}\right)=\frac{1}{5} \times 0.25=0.05 .$ Hence $\mathrm{P}(A)=0.75+0.05=0.8$.
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