A knight$-$errant seeking to slay a dragon arrives at a fork with a road sign saying that, on the leftmost path, he faces a $0\mathrm{.}6$ probability of meeting Grendel the Dragon, who has a record of killing $70\%$ of the knights who fight him; on the middle path, there is a $0\mathrm{.}5$ probability of meeting Smaug the Dragon, who kills $60\%$ of the contenders; and on the rightmost path, there is a $0\mathrm{.}4$ probability of meeting Puffie the Dragon, who kills $30\%$ of the contenders. The knight decides to trust his fate and to cast a die: $1$ or $2$ will send him on the road to Grendel, $3$ or $4$ to Smaug, and $5$ or $6$ to Puffie. $($Don$$t forget this initial choice of path when you set up a diagram.$)$

$($a$)$ What are his chances to come to the princess alive? $($Remember that this will happen if he either doesn$$t meet a dragon $$ or he does, but the dragon fails to kill him.$)$

$($b$)$ If he shows up at his date, how likely is it that he followed the rightmost path?