Let $E$ and $F$ be events with $P(E)=\frac{3}{5}, P(F)=\frac{3}{10}$ and $P(E \cap F)=\frac{1}{5}$. Are $E$ and $F$ independent?

If $A$ and $B$ are two non-mutually exclusive events such that $P(A \mid B) = P(B \mid A)$,then

If $P(A) = \frac{3}{10}$,$P(B) = \frac{3}{5}$,and $P(A \cup B) = \frac{3}{5}$,then find the value of $P(A|B) \times P(B|A)$.

$A$ bag contains $N$ balls out of which $3$ are white,$6$ are green,and the remaining $(N-9)$ balls are blue. Three balls are drawn randomly one after the other without replacement. Let $W_i, G_i$,and $B_i$ denote the events that the ball drawn in the $i^{\text{th}}$ draw is white,green,and blue,respectively. If $P(W_1 \cap G_2 \cap B_3) = \frac{2}{5N}$ and $P(B_3 \mid W_1 \cap G_2) = \frac{2}{9}$,then $N$ equals:

From a lot containing $n$ good and $m$ bad articles, if $2$ articles are picked at random in succession without replacement, then the probability that the second article picked is bad is

One ticket is selected at random from $100$ tickets numbered $00, 01, 02, \dots, 98, 99$. If $X$ and $Y$ denote the sum and the product of the digits on the tickets,then $P(X = 9 | Y = 0)$ equals