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Math puzzler of the week

Edgy DC Jul 30 2007 01:53 PM

It was once stated, that at the height of the Beatles popularity, popular music stations played their music 40% of the time. Assuming that's true, f you tuned through 10 such stations at any given moment, what is the probability that at least one of the stations would be playing a Beatles song?

DocTee Jul 30 2007 01:56 PM

40%

DocTee Jul 30 2007 01:56 PM

Do I need to show my work?

Edgy DC Jul 30 2007 01:58 PM

No. But marathon would.

Benjamin Grimm Jul 30 2007 01:59 PM

Here's my approach: For each station, there's a 0.6 chance that it's NOT playing a Beatles song. So the odds of all 10 stations NOT playing the Beatles at any given moment are 0.6 * 0.6 ... ten times. Which is 0.0060466176. That means that there would be a 99.9939533824 per cent chance that you'd find a Beatles song on one of the ten stations. How's that sound? Did I do it right?

Edgy DC Jul 30 2007 02:07 PM

Move over, Reed Richards. Ben Grimm's 99.39533824% is correct. Show him what he wins, Sue.

metsmarathon Jul 30 2007 02:12 PM

without much/any additional thought, i'd be inclined to think that yancy has it right. well, sorta. is it just 40% of music played (4 songs in 10), 40% of all songplay (4 minutes for every 10 minutes of music), or 40% of all airtime (9.6 hours per day)? cos if its anything but the latter, then there's probably a decent chance that you'd find all ten radio stations on a commercial break, or between programs, at certain times of the day, like 6 am, or 2 pm, or noon or whatever. but that aside, i'd lean towards the near 100% number yancy gave.

Edgy DC Jul 30 2007 02:22 PM

played their music 40% of the time. I think, for the purposes of our question, the implication is total air time. The Thing was right to the last decimal.

DocTee Jul 30 2007 02:25 PM

Huh? Math was never my strong suit, but if the chance of a station playing a Beatles tune is 40%, then you have a 40% chance of hearing one, no? So that' a 40% chance for each of these ten stations. Meaning if you tuned to anyone of them, there is a 40%... never mind I give up

Edgy DC Jul 30 2007 02:27 PM

You're tuning through ten stations.

metirish Jul 30 2007 02:29 PM

40% X 10..........400%?

Benjamin Grimm Jul 30 2007 02:31 PM

DocTee wrote: Huh? Math was never my strong suit, but if the chance of a station playing a Beatles tune is 40%, then you have a 40% chance of hearing one, no? So that' a 40% chance for each of these ten stations. Meaning if you tuned to anyone of them, there is a 40%... never mind I give up Let's say that 50% of the people in this world are female. If there are ten random people in a room, what are the chances that at least one is female? Certainly more than 50%. It's actually, by the method I used above, 99.9990234375 per cent.

DocTee Jul 30 2007 02:31 PM

got it.

Frayed Knot Jul 30 2007 02:38 PM

Think of it like a coin-flip situation. For NOT ONE female to walk into the room would be like the coin turning up heads 10 times in a row. Long odds indeed. The Beatles music is the same idea except that the "coin" in this case only comes up heads 4 times in 10 rather than 5. So to miss heads all ten flips is still almost impossible.

HahnSolo Jul 30 2007 02:43 PM

I say we send it to Ask Marilyn. I'm still closer to DocTee's original thinking.

DocTee Jul 30 2007 02:44 PM Edited 1 time(s), most recently on Jul 30 2007 02:51 PM

My thinking was along the lines of a coin-flip: turn to ten stations and you have a 40% chance of hearing a Beatles tune on any one of them. The chances don't increase/decrease based on what was heard on previous stops on the dial. It's 40% on each of the stations...I guess I mis-read the question, or mis-understood it's complexity.

HahnSolo Jul 30 2007 02:48 PM

I submitted this online to Ask Marilyn. I'll see if I get a response.

HahnSolo Jul 30 2007 07:22 PM

="DocTee"]My thinking was along the lines of a coin-flip: turn to ten stations and you have a 40% chance of hearing a Beatles tune on any one of them. The chances don't increase/decrease based on what was heard on previous stops on the dial. It's 40% on each of the stations...I guess I mis-read the question, or mis-understood it's complexity. I believe the answer is 40%, and I think the bolded part is important. For instance, if you checked nine stations and none were playing a Beatles song, there's still only a 4 in 10 chance that the tenth station is playing one. In 1941, Ted Williams hit .406, which means he got four hits every 10 at bats (just about). And if I chose 10 random of his at bats from that year, what are the odds that one of them would be a hit? It would only be 4 out of 10.

DocTee Jul 30 2007 07:38 PM

]Think of it like a coin-flip situation. For NOT ONE female to walk into the room would be like the coin turning up heads 10 times in a row. Long odds indeed. The Beatles music is the same idea except that the "coin" in this case only comes up heads 4 times in 10 rather than 5. So to miss heads all ten flips is still almost impossible. If heads comes up a million times in a row, the chances of it being tails the next time the coin is flipped is still "only" 50%.

Edgy DC Jul 30 2007 07:42 PM

We're talking about the odds of a non-Beatles song the next time. We're talking about the odds of a non-Beatles song ten times in a row.

Benjamin Grimm Jul 31 2007 05:16 AM

A lot of people have trouble understanding the basic laws of probability. That's why they're able to sell lottery tickets.

metsmarathon Jul 31 2007 07:34 AM

lets say your tasked with flipping a coin until it comes up tails. on the first flip, there is a 0.50 probability that it will turn up heads, and a 0.50 probability that it will turn up tails. if it turns up tails, you're done. if it turns up heads, you have to flip it again. so there's a 0.50 probability that you will flip the coin exactly once, and a 0.50 probability that you will flip the coin at least twice. if you flip the coin the second time, you have a 50% probability that on this flip, it will come up heads, and a 50% probability that it will come up tails. if it comes up tails, you have the 0.50 probability that you've needed to flip the coin the second time, multiplied by the 0.50 probability that on the second flip, it came up tails. therefore, in your task, you have the following: 0.50 probability that you flip exactly once 0.50 x 0.50 = 0.25 probability that you flip exactly twice and here's the important part - the sum of all likelihoods must equal 1. no matter what the task, the sum the probabilities of all possible outcomes of that task MUST be exactly equal to 1. and so, if you're flipping a coin, and you didn't get tails on the first flip, and you didn't get tails on the second flip, then the probability that you will need to flip that coin at least three times is equal to 1 minus the probability that you will have already stopped flipping the coin, or 1 - (0.50 + 0.25) = 1 - 0.75 = 0.25 so there's a 0.25, or 25% chance that you will need to flip a coin at least three times before it comes up tails. and on that third flip, there is again a 0.50 probability that it will turn up heads, meaning there is a 0.25 x 0.50 = 0.125 or 12.5% chance that you will need to flip the coin at least four times. so lets get into some algebra here... for any flip, the probability that your coin flip will come up tails, P(t) = 0.5 P stands for probability. the t in the parentheses merely stands for the outcome you're looking for. so, for the probability that you require one flip to get tails, P(t) = 0.50 and for the probability that you require two flips to get tails, P(t) = 0.50 x 0.50, or P(t) = 0.50 ^ 2 = 0.25 and for the probability that you require three flips to get tails, P(t) = 0.50 x 0.50 x 0.50, or P(t) = 0.50 ^ 3 = 0.125 so if we continue this out, we see the following: fourth flips: 0.50 ^ 4 = 0.0625 fifth flips: 0.50 ^ 5 = 0.03125 sixth flips: 0.50 ^ 6 = 0.015625 seventh flips: 0.50 ^ 7 = 0.0078125 eighth flips: 0.50 ^ 8 = 0.00390625 ninth flips: 0.50 ^ 9 = 0.001953125 tenth flips: 0.50 ^ 10 = 0.0009765625 and the probability that it will require more than ten flips is equal to the probability that it will require exactly ten flips, 0.50 ^ 10 = 0.0009765625 and therefore, the probability that you can flip a coin ten times, and have it come out heads each time, is 0.50 ^ 10 = 0.0009765625. and in other words, there is a 99.90234375% chance that the coin will turn up tails at least once. if we then extend this analogy to the radio station, there is a 0.40 probability that any one station is playing hte beatles, and a 0.60 probability htat any one station is NOT playing the beatles. and so, you turn on the radio. and if the first station is playing the beatles, you stay on that station, because you're a beatles fan. but there's a 60% chance that they will not be playing a bealtes song, and being the rabid beatles fan you are, you turn the dial hoping for another beatles song. on that second station, there is again a 40% chance that they will be playing a beatles song, but there's still that 60% chance that they will not be playing a beatles song. and going back to the math of it, P(b) = 0.40, or the probability of a beatles song is 0.40. P(o) = 0.60, or the probability of some "other" song is 0.60 the probability that the first station you turn to is not playing a beatles song, as we've said, is 0.60, P(o) = 0.60 the probability that the second station is not playing a beatles song is also 0.60. but the probability that BOTH stations are not playing a beatles song is 0.60 x 0.60 = 0.36 and the probability that the third station you turn to is ALSO not playing the beatles is 0.60 x 0.60 x 0.60, or 0.60 ^ 3 = 0.216 and carrying this out quite a ways... four stations: 0.60 ^ 4 = 0.1296 five stations: 0.60 ^ 5 = 0.07776 six stations: 0.60 ^ 6 = 0.046656 seven stations: 0.60 ^ 7 = 0.0279936 eight stations: 0.60 ^ 8 = 0.01679616 nine stations: 0.60 ^ 9 = 0.010077696 ten stations: 0.60 ^ 10 = 0.0060466176 the probability that ten stations will not, all at the same time, be playing a beatles song, is 0.0060466176, or 0.60466176% and since the sum of all possible outcomes is 1, the probability that AT LEAST ONE of these ten stations is playing a beatles song is equal to 1 minus the probability that NONE of them are playing a beatles song: 1 - 0.0060466176, or 0.9939533824 as winnie cooper might say, [url=http://www.cnn.com/2007/SHOWBIZ/TV/07/30/people.mckellar.ap/index.html]math doesn't suck.[/url]

Edgy DC Jul 31 2007 07:50 AM

Winnie gave me the problem. I'll go to Cindy Brady or Malory Keaton or Tootie Ramsey or somebody for next week's puzzler.

Johnny Dickshot Jul 31 2007 08:12 AM

I flipped a quarter 10 times while reading mm's post. Here are the results: T H T T H T H H H H This proves, without a doubt, that Winnie Cooper is Hot.

Benjamin Grimm Jul 31 2007 08:35 AM

Confession: until I just looked her up, I had no idea who Winnie Cooper was.

m.e.t.b.o.t. Jul 31 2007 06:20 PM

m.e.t.b.o.t. agrees with human actress danica mckellar, but offers no insight into her average kinetic energy.