Amongst all vehicles that have an ignition problem, assume that the starter is at fault in $60\%$ of the cases; the battery is at fault in $30\%$ of the cases, and one of the plugs is at fault the rest of the time. Furthermore, amongst all vehicles with an ignition problem, the proportion of vehicles of make X is $5\%$ in cases where the starter is at fault; $20\%$ in cases where the battery is at fault, and $10\%$ in cases where one of the plugs is at fault.

$1.$ A vehicle of make X is brought to a mechanic because of an ignition problem. What should be the first line of attack for the mechanic: starter, battery, or plugs?

$2.$ A vehicle of make X is brought to a mechanic because of an ignition problem. What is the probability that the starter is not at fault? Solve this question using two approaches