Two dice are thrown Defined are the following two events

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Question 8:- Two dice are thrown Defined are the following two events A and B :  A = {(x, y): x + y = 9}, B = {(x, y) : x ≠ 3}, Where (x, y) denote a point in the same sample space.

Check if events A and B are independent or mutually exclusive.

Solution:-  Since, Two dice are thrown

Then, A = {(x, y): x + y = 9}

A = {(3, 6) (4, 5) (5, 4) (6, 3)}

B = {(x, y) : x ≠ 3}

B = {(1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6) (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6) (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6) (5, 1) (5,2) (5, 3) (5, 4) (5, 5) (5, 6) (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6)}

A∩B = {(4, 5) (5, 4) (6, 3)}

P(A) = 4/36, P(B) = 30/36

P(A).P(B) = \dfrac{4}{36}.\dfrac{30}{36}= \dfrac{5}{36}

P(A∩B) = 3/36

Since, P(A∩B) ≠ P(A).P(B) and P(A∩B) ≠ 0

Hence, A and B are not independent or mutually exclusive

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(1+x^2)\dfrac{dy}{dx} + 2xy = 4x^2.

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Solution:- See full solution

Question 5:- Find the shortest distance between the lines :

\dfrac{x + 1}{2} = \dfrac{y - 1}{1} = \dfrac{z - 9}{-3}

\dfrac{x - 3}{2} = \dfrac{y + 15}{-7} = \dfrac{z - 9}{5}.

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