Calculate the area of the shaded region
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The song of the day is announced at a random time between 7:00 and 7:30, so you have a 30-minutes span of time where the song could be announced. If you start listening to the radio 20 minutes in, you're at 2/3 of the time. So, there is a 2/3 chance that the song has already been chosen, and a 1/3 chance that it will be announced in the remaining 10 minutes.
Similarly, the trivia question is asked at a random time between 7:15 and 7:45 so there is again a 30-minutes span of time. But this time, we're only 5 minutes late, so there is a 5/30=1/6 chance that the question was already asked, and a 25/30=5/6 chance that the question will be asked in the remaining 25 minutes.
So, there are four scenarios:
- You have missed both the song of the day and the trivia question. Given all we said, this event has a chance of
- You miss the song, but not the trivia:
- You miss the trivia, but not the song
- You don't miss anything:
You're interested in the probability of missing either of the announcements, so you want to sum the two middle cases (in the first you miss everything, in the second you don't miss anything).
Your result is thus