I'm playing a game. I enter three different modes at different probability.
For the first one, the probability of entering is $\dfrac17$ and that of winning is $\dfrac23$. For the second one, the probability of entering is $\dfrac27$ and that of winning is $\dfrac12$. For the first one, the probability of entering is $\dfrac47$ and that of winning is $0$.
Find the probability of my victory.
Method $1$: $$P=\frac17\times\frac23+\frac27\times\frac12+\frac47\times0=\frac5{21}.$$
Method $2$: Draw a tree like this.
I use the tick and cross to represent victory and defeat. And to incarnate different probability, I repeatedly draw. From the figure, $P=\dfrac6{13}\ne\dfrac8{21}$.
So the two methods lead to different answers. Although this has happened in probability like here, I still suppose that one of the solutions is wrong.