Can't Sleep

There's a digital alarm clock that displays 24-hour time by using $3$ or $4$ digits depending on the time of the day. Every day the clock starts at $0:00$ , makes its way to $9:59$ , then goes to $10:00$ , along to $19:59$ , then displays $20:00$ , and then finishes the day at $23:59$ . The clock never displays $24:00$ .

Suppose that once a day for $50$ days, you check the clock at a completely random time, and select one of the shown digits randomly to record in your journal.

What is the expected number of days that an even digit was chosen and recorded in your journal, to one decimal place?

Note: $0$ is considered an even number.