Please submit solutions to the following problems. Be sure to show all your work.

Many clocks display numbers using arrays like the following.

The device lights up different parts of the array $($any segment lettered a through g$),$ to display any single digit $0$ through $9.$ Suppose a digit is chosen at random. Determine the probability for each of the following. $($You can use a digital clock to see which segments are lit up for the different numbers.$)$

$($a$)$ Segment a will be lit.

$($b$)$ Segment b will be lit.

$($c$)$ Segments $e$ and $b$ will both be lit.

$($d$)$ At least one of segments e or b will be lit.