A student appears for tests I,II,and III. The | vedclass

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(A) Let $X$ be the event that the student is successful. Let $X_1, X_2, X_3$ be the events of passing tests $I, II, III$ respectively.Given $P(X_1) =

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$p(1+q)=1$

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(A) Let $X$ be the event that the student is successful. Let $X_1, X_2, X_3$ be the events of passing tests $I, II, III$ respectively.Given $P(X_1) = p$,$P(X_2) = q$,and $P(X_3) = 1/2$.The student is successful if $(X_1 \cap X_2)$ or $(X_1 \cap X_3)$.Since these events are not mutually exclusive,we use the formula $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.Here,$A = X_1 \cap X_2$ and $B = X_1 \cap X_3$.$P(X) = P(X_1 \cap X_2) + P(X_1 \cap X_3) - P(X_1 \cap X_2 \cap X_3)$.Since the tests are independent:$P(X) = p \cdot q + p \cdot (1/2) - p \cdot q \cdot (1/2)$.Given $P(X) = 1/2$:$1/2 = pq + p/2 - pq/2$.$1/2 = p/2 + pq/2$.Multiply by $2$:$1 = p + pq$.$1 = p(1+q)$.

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