GATE | GATE CS 1999 | Question 26

Consider two events E1 and E2 such that probability of E1, Pr[E1]=1/2, probability of E2, Pr[E2]=1/3, and probability of E1 and E2, Pr[E1 and E2]=1/5, Which of the following statement is/are True?
(A) Pr[E1 and E2] is 2/3
(B) Events E1 and E2 are independent.
(C) Events E1 and E2 are not independent.
(D) Pr[E1/E2]= 4/5

Answer: (C)

Explanation: Given-
$P(E1) = \frac{1}{2}$
$P(E2) = \frac{1}{3}$
$P(E1 \cap E2) = \frac{1}{5}$
Therefore, $P(E1 \cup E2) = \frac{1}{2} + \frac{1}{3} - \frac{1}{5} = \frac{19}{30}$
$P(E1|E2) = \frac{1/5}{1/3} = \frac{3}{5}$
A option is wrong, since it is given in the question.
B option is wrong since $P(E1|E2)$ is not equal to $P(E1 \cap E2)$
D option is wrong since $P(E1|E2) = \frac{3}{5}$
Therefore C option is correct.

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