Are the events independent?

Three zinked coins are given. The probabilities for the head are $ \frac{2}{5} $, $ \frac{3}{5} $ and $ \frac{4}{5} $. A coin is randomly selected and then thrown twice. $ M_k $ denotes the event that the $ k $th coin was chosen with $ k = 1,2,3 $. $ K_j $ stands for the event that at the $j$th throw we get head, where $ j = 1,2 $.

I want to calculate the probability $P(K_2\mid K_1)$.

From the definition of conditional probability we get that $P(K_2\mid K_1)=\frac{P(K_2\cap K_1)}{P(K_1)}$.

Are the events $K_1$ and $K_2$ independent? Does it hold that $P(K_2\cap K_1)=P(K_2)\cdot P( K_1)$ ?

Solutions Collecting From Web of "Are the events independent?"

The events are dependent because getting H on first toss gives more evidence that $M_3$ happened, making it more likely that I would get second head as well.

Not independent:
$$
P(K_1\land K_2)=\frac13\frac25\frac25+\frac13\frac35\frac35+\frac13\frac45\frac45=\frac{29}{75}
$$
and
$$
P(K_j)=\frac13\frac25+\frac13\frac35+\frac13\frac45=\frac35
$$