I play a little game with myself while listening to Sirius radio. When a song comes on the 60’s or 70’s channel, it displays the artist, the title and the year the song was released. I try to guess the year. I get the answer “correct” if I’m not off by more than one year. EG: If the song was released in 1965 and I guessed 1964, I’ll give it to myself. Since the 60’s channel only plays 60’s music, I know that the answer must be from 1960 - 1969.
Now, assuming that I knew absolutely nothing about 60’s music, one could reasonably assume that my success rate would be 3 in 10. However, since I give myself credit for only missing it by one year, I will never guess the 2 extremes of 1960 or 1969 since those would limit my chances of guessing correctly to 1959 (which should be impossible), 1960 or 1961 and of course the same is true for guessing 1969.
How do these 2 outliers affect my odds? I assume that they’re greater than 3 in 10 but I don’t know how much.
Your guesses are 61-68, eight different guesses. The possible correct answers are 60-69, ten answers. Assuming your guesses are random, the odds of a correct answer is the sum of the odds of each of them being correct, divided by 8. Now since each possible guess has 3 in ten odds, the result is 3 in ten.
And I just made myself go :smack: d’oh! Of course! You could be just answering 1965 all the time, and your odds would be 3 in 10. The odds only change if you include your outliers, then it’s (83/10 + 22/10) in ten. 2.8 in ten.
Not to rain on your parade, but satellite radio’s “decade” stations definitely overlap. You’ll get some stuff from '59 and '70-'71 on the “60s” station, and similarly on the other stations. They don’t have a 2000s station, so the 90s on 9 creeps into 2001-2002 era. 40s on 4 was my favorite, because they’d play stuff from the 20s all up through the early 50s.
No rain at all. I’d throw those out, although I don’t actually remember seeing any from “out of decade” except when Cousin Brucie comes on and takes requests.
Assuming there are equal numbers of songs from each year, and you’re choosing randomly (from 1961 to 1968), then odds are 3 in 10.
You’re only guessing from 8 years, but if the song is actually 1960, only one year wins for you, and if the song is 1961, only two years win. These cancel out, and the odds of winning is 3/10.
Enumerate every S,G pair (song, guess). Assuming every year song and guess is equally likely, and random, one does not influence the other, then it’s basically 3 in 10. (24 in 80)
It makes no difference if the guesses are equally likely as long as the songs are. As previously pointed out, you could guess 1965 for everything and still get 3 in 10 right.
The same is not true if the guesses are equally likely but the songs are not. For example if all songs are all from 1960 then equally likely guesses (from '61 to '68) would produce only 1 in 8 correct. Similarly if '60 and '69 songs were more common than the others, you’d score less than 3 in 10 with random guessing. I believe than songs would have to be twice as likely for each year '61 to '68 as for '60 or '69 to make equally likely guessing ('61 to '68) to be the optimal random strategy.
If the songs aren’t uniformly distributed, then your optimum strategy is to always guess whichever year they’re clustered around. If they are uniformly distributed, then any strategy which doesn’t pick the endpoints is equally good. The only case where uniformly-random guessing will actually help you is the case where someone at the station is directly competing against you and trying to make you lose.
If the guess is truly random (like, coming from a random number generator, dice, et cetera) then it doesn’t matter whether you hear the song before or after the guess. So turn the problem around. Make the guess first, then find out what the song is and what year it’s from.
There are eight possible guesses. For every single one of those possibilities, there are exactly ten possibilities of what year the song will be from, and exactly three of those years will be within one year of the guess. So the probability of winning with random guessing is 3/10 or 30%. If you play the game and try to win using your knowledge of music, you should be winning more than 30% of the time.
But you asked what are the ODDS.
3/10 you win, 7/10 you lose, so the odds are 7 to 3 against winning by guessing. So, if you want to set up a fair game for people to play at the Fall Carnival, the payout ratio should be 7 to 3. It should cost $3 to play the game and when you win you get $7. If your goal is to rig the game so that, in the long run, the house makes a profit, you need to set the winning ratio at lower than 7 to 3. For example, you could charge $1 to play and pay out $2 when they win. 2/1 is lower than 7/3 so the house makes a profit, assuming that most players have very little knowledge about 1960s music.