Final answer:
The probability of missing either the song of the day announcement or the trivia question is 5/6. This considers the periods before the student starts listening at 7:20 a.m. for both events, which are independent and non-overlapping.
Step-by-step explanation:
To calculate the probability of missing either the song of the day announcement or the trivia question, we need to consider the timing of both events and when you begin listening to the station.
The song of the day is played at a random time between 7:00 and 7:30 a.m., this is a half-hour window. If you start listening at 7:20 a.m., you may miss the song of the day if it plays in the 20 minutes before you start listening, which is between 7:00 and 7:20 a.m.
The trivia question is asked at a random time between 7:15 and 7:45 a.m., a half-hour span. Since you begin listening at 7:20 a.m., you could miss the trivia question if it is asked in the 5 minutes from 7:15 to 7:20 a.m.
So, the probability of missing the song is the 20-minute window you're not listening out of the 30-minute total, which is 20/30 or 2/3. The probability of missing the trivia question is the 5-minute window out of the 30-minute total, which is 5/30 or 1/6.
To find the probability of missing at least one of the events, we need to add the probabilities together, considering they are independent and non-overlapping events:
P(missing song or trivia) = P(missing song) + P(missing trivia)
P(missing song or trivia) = 2/3 + 1/6 = 4/6 + 1/6 = 5/6.
This is because you are already listening during the overlap time (7:20 to 7:30) for both events.