Respuesta :
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
[tex]\dfrac{2}{3}\cdot\dfrac{1}{6} =\dfrac{2}{18}[/tex]
- You miss the song, but not the trivia:
[tex]\dfrac{2}{3}\cdot\dfrac{5}{6} =\dfrac{10}{18}[/tex]
- You miss the trivia, but not the song
[tex]\dfrac{1}{3}\cdot\dfrac{1}{6} =\dfrac{1}{18}[/tex]
- You don't miss anything:
[tex]\dfrac{1}{3}\cdot\dfrac{5}{6} =\dfrac{5}{18}[/tex]
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
[tex]\dfrac{10}{18}+\dfrac{1}{18} = \dfrac{11}{18}[/tex]
The probability of missing out on any one of the events is [tex]\frac{11}{18}[/tex]
What is probability?
'Probability can be defined as the ratio of the number of favorable outcomes to the total number of outcomes of an event.'
According to the given problem,
Within 30 minutes time span, when arrived 20 minutes late, probability of the song already been chosen = [tex]\frac{20}{30}[/tex]
= [tex]\frac{2}{3}[/tex]
Probability of the song to be chosen in the remaining 10 minutes = [tex]\frac{10}{30}[/tex]
= [tex]\frac{1}{3}[/tex]
Now,
within 30 minute time span, when arrived 5 minutes late, probability of the trivia question already been asked = [tex]\frac{5}{30}[/tex] = [tex]\frac{1}{6}[/tex]
Probability of the question to be asked in the remaining 25 minutes = [tex]\frac{25}{30}[/tex]
= [tex]\frac{5}{6}[/tex]
Probability of missing out on both the events = [tex]\frac{2}{3} * \frac{1}{6}[/tex]
= [tex]\frac{1}{9}[/tex]
Probability of missing out song but not the trivia question = [tex]\frac{2}{3} *\frac{5}{6}[/tex] = [tex]\frac{5}{9}[/tex]
Probability of missing out on trivia question but not the song = [tex]\frac{1}{3} *\frac{1}{6}[/tex] = [tex]\frac{1}{18}[/tex]
Therefore, the probability on missing out any one of the events = [tex]\frac{5}{9} + \frac{1}{18}[/tex]
= [tex]\frac{11}{18}[/tex]
Hence, we can conclude that there is [tex]\frac{11}{18}[/tex] chances of missing out on any one of the events.
Learn more about probability here: https://brainly.in/question/34187875
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