question_answer

A fair die is rolled. Consider the following events A = {2, 4, 6}, B= {4, 5} and C={3, 4, 5, 6}.Find (i) \[P(A\cup B/C)\] (ii) \[P(A\cap B/C).\]

Answer:

Given events are A={2, 4, 6}, 8 = {4, 5} and         C = {3, 4, 5, 6} Sample space, S = {1, 2, 3, 4, 5, 6} Now, \[A\cup B=\] {2, 4, 5, 6}, \[A\cap B=\]{4} \[A\cup B\,\cap C=\]{2, 4, 5, 6} \[\cap \] {3, 4, 5, 6} = {4, 5, 6} and \[A\cup B\,\cap C=\] {4} \[\cap \] {3, 4, 5, 6} = {4} \[\therefore \] \[n(S)=6,(A\cup B\cap C)=3,\]\[n(A\cap B\cap C)=1\] and n(C) = 4 (i) \[P(A\cup B/C)=\frac{P(A\cup B\cap C)}{P(C)}\] \[=\frac{n(A\cup B\cap C)/n(S)}{n(C)/n(S)}\]             \[=\frac{3/6}{4/6}=\frac{3}{4}\] (ii) \[P(A\cap B/C)=\frac{P(A\cap B\cap C)}{P(C)}\] \[=\frac{n(A\cap B\cap C)/n(S)}{n(C)/n(S)}\]             \[=\frac{1/6}{4/6}=\frac{1}{4}\]