# 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
$$P(K_j)=\frac13\frac25+\frac13\frac35+\frac13\frac45=\frac35$$