Indy 31.4 is a radio station that plays music 24 hours a day, 7 days a week. They have edited every song they play so that the duration of the song is exactly 3 minutes. The producers of Indy 31.4 decided that the radio station should play music non-stop so that as soon as one song has completed, the next song will be played immediately. To cater to their listeners, Indy 31.4 will allow listeners to call in and request songs. However, a request can only be made for the next song played. So a listener must call in during the 3 minutes of a song to make a request for the next one. If multiple users make a request, only one of them (chosen randomly with equal probability) will be played and the rest of the requests are thrown out. If no calls arrive during a song, the DJ will choose the next song. Calls arrive following a Poisson process with rate ? = 0.5 per minute.
a) What is the probability that a song being played is one that is requested by a listener?
b) What is the probability that there are exactly two requests for the next song?
c) What is the probability that the next song will have at least 3 requests?
d) If exactly two calls arrive during a song, what is the probability that both calls arrived during the first minute of the song?
e) Assume the rule is changed so that requests are not thrown out, but are added to a queue and played in the order of their requests. If a song just started playing and was a song chosen by the DJ, what is the probability that the song after the next song (not the current one, not the next one, the one after that) is also chosen by the DJ?