amazing bad luck
by John at 12/01/2004 04:13:00 PM
Last night on "Amazing Race", one of the tasks (I think it is called a Road Block) was to find one of 20 "Amazing Race" clues hidden inside 200 lb rolls of hay in a field with 240 rolls of hay.
The odds of picking a hay bale with a clue in it on the first try: 1 in 12.
The odds of picking through 100 hay bales and never finding a clue: about 17 in 100000!
That's just what happened, though. One member of the sister team unrolled hay bales for about eight hours, and the host estimated that she personally unrolled 100 of the 240 hay bales in the field. Either she just kept missing the clues (which were large, yellow and black envelopes about the size of a sheet of copy paper) as she pulled apart the bales, or she was really unfortunate.
She wasn't giving up, and eventually the host stepped in and brought the whole thing to a halt because their team was going to be eliminated from the race anyway. "A" for effort, though.
Update: In the car on the way home, I realized that my first approximation was not very accurate. More realistic odds are anywhere from about 7-15 times worse.
Of course, the sister will not look at the same hay bale more than once - this reduces her odds of not finding a clue in 100 bales to about 13 in 1 million. Mitigating this is the fact that 8 other teams were also looking for clues. On average, a team would have dig through 8 bales of hay to have even odds of finding a clue, so the other teams will look through 64 bales. To give her the best possible odds, we assume the other teams search through 64 bales and find 8 clues before she even starts (it didn't actually work this way - several teams search concurrently, and the sisters were one of the first teams to start). Her odds of not finding a clue in 100 bales is now about 25 in 1 million. Pretty bad luck!
Update 2: I can't seem to let it go. In the shower this morning (12/2), it occurred to me that my previous estimate of an average of 8 bales of hay per team to find a clue was wrong. I think it is closer to 12 bales per team, but again, this ignores the fact that teams won't look in bales that have already been unrolled. Anyway, at an average of 12 bales of hay per team for the first 8 teams, the odds of the sisters unrolling another 100 bales without finding a clue go to about 2 in 10 million!
And finally, I settled on a way to get this out of my system - stochastic modeling. I couldn't really stochasticly simulate the very long odds against unrolling 100 hay bales without finding a clue (actually, I could, but it would take a long time), so instead I simulated the number of bales for each of the other 8 teams to find a clue. It takes, on average, about 92 bales for the other 8 teams to find 8 clues. This means that the odds of the sisters team unrolling a further 100 bales without finding a clue are about 5 in 10 million. Still pretty long odds.
Update 3: Why are you still reading this? If you are as fascinated by this problem as I am, I feel sorry for you.
So, I caved. I made a full stochastic model of the Road Block. There are nine teams, and they each pick a bale of hay and unroll it looking for a clue in turn. If they find one, they stop; if not they pick another unrolled bale. I kept track of how many bales a team has to unroll before it finds a clue. This game is repeated 100,000 times, and it only takes about 2.5 minutes to play all of them.
Like I said above, the mean number is about 12 bales per team - on average, it takes 11.48+-5.27 bales for each team to find a clue (95% confidence interval). In every game, of course, one team has to unroll more bales than the other teams. The average maximum number unrolled is 31.55+-22.56. The big question: how many times in 100,000 games did a team need more than 100 tries before they found a clue? The big answer: 9 times. To bracket that with a standard deviation, I'd have to run the 100,000 game simulation a bunch of times (one rule of thumb for standard deviation is about 25 samples), which I think will take about an hour. I'll get right on it.
Anyway, It's probably safe to say that the odds against the sisters team having to unroll 100 hay bales are about 1 in 10,000. This makes my previous analysis look faulty, but I think that can be explained by the constraints I put on the game (specifically, the sisters only get to unroll hay bales after everyone else has found a clue). This model is more realistic, but in the actual event, some teams started earlier than others and finished before the other teams even began, so it is still not a very accurate recreation of the show.
Update 4: 25 simulations later...
The odds: 8.72+-4.10 in 100,000. Or probability 0.00872+-0.00410%. Or (the way Wendy likes it) 1 chance in 11468 (21,645 - 7,800).
The odds of picking a hay bale with a clue in it on the first try: 1 in 12.
The odds of picking through 100 hay bales and never finding a clue: about 17 in 100000!
That's just what happened, though. One member of the sister team unrolled hay bales for about eight hours, and the host estimated that she personally unrolled 100 of the 240 hay bales in the field. Either she just kept missing the clues (which were large, yellow and black envelopes about the size of a sheet of copy paper) as she pulled apart the bales, or she was really unfortunate.
She wasn't giving up, and eventually the host stepped in and brought the whole thing to a halt because their team was going to be eliminated from the race anyway. "A" for effort, though.
Update: In the car on the way home, I realized that my first approximation was not very accurate. More realistic odds are anywhere from about 7-15 times worse.
Of course, the sister will not look at the same hay bale more than once - this reduces her odds of not finding a clue in 100 bales to about 13 in 1 million. Mitigating this is the fact that 8 other teams were also looking for clues. On average, a team would have dig through 8 bales of hay to have even odds of finding a clue, so the other teams will look through 64 bales. To give her the best possible odds, we assume the other teams search through 64 bales and find 8 clues before she even starts (it didn't actually work this way - several teams search concurrently, and the sisters were one of the first teams to start). Her odds of not finding a clue in 100 bales is now about 25 in 1 million. Pretty bad luck!
Update 2: I can't seem to let it go. In the shower this morning (12/2), it occurred to me that my previous estimate of an average of 8 bales of hay per team to find a clue was wrong. I think it is closer to 12 bales per team, but again, this ignores the fact that teams won't look in bales that have already been unrolled. Anyway, at an average of 12 bales of hay per team for the first 8 teams, the odds of the sisters unrolling another 100 bales without finding a clue go to about 2 in 10 million!
And finally, I settled on a way to get this out of my system - stochastic modeling. I couldn't really stochasticly simulate the very long odds against unrolling 100 hay bales without finding a clue (actually, I could, but it would take a long time), so instead I simulated the number of bales for each of the other 8 teams to find a clue. It takes, on average, about 92 bales for the other 8 teams to find 8 clues. This means that the odds of the sisters team unrolling a further 100 bales without finding a clue are about 5 in 10 million. Still pretty long odds.
Update 3: Why are you still reading this? If you are as fascinated by this problem as I am, I feel sorry for you.
So, I caved. I made a full stochastic model of the Road Block. There are nine teams, and they each pick a bale of hay and unroll it looking for a clue in turn. If they find one, they stop; if not they pick another unrolled bale. I kept track of how many bales a team has to unroll before it finds a clue. This game is repeated 100,000 times, and it only takes about 2.5 minutes to play all of them.
Like I said above, the mean number is about 12 bales per team - on average, it takes 11.48+-5.27 bales for each team to find a clue (95% confidence interval). In every game, of course, one team has to unroll more bales than the other teams. The average maximum number unrolled is 31.55+-22.56. The big question: how many times in 100,000 games did a team need more than 100 tries before they found a clue? The big answer: 9 times. To bracket that with a standard deviation, I'd have to run the 100,000 game simulation a bunch of times (one rule of thumb for standard deviation is about 25 samples), which I think will take about an hour. I'll get right on it.
Anyway, It's probably safe to say that the odds against the sisters team having to unroll 100 hay bales are about 1 in 10,000. This makes my previous analysis look faulty, but I think that can be explained by the constraints I put on the game (specifically, the sisters only get to unroll hay bales after everyone else has found a clue). This model is more realistic, but in the actual event, some teams started earlier than others and finished before the other teams even began, so it is still not a very accurate recreation of the show.
Update 4: 25 simulations later...
The odds: 8.72+-4.10 in 100,000. Or probability 0.00872+-0.00410%. Or (the way Wendy likes it) 1 chance in 11468 (21,645 - 7,800).
I just wasn't interested. I saw the trailer a couple of times, and heard an interview on NPR with the novel/screenplay writer.
Why wasn't I interested? I didn't want to hear people obsess over wine for 2 hours - I think that would probably annoy me. And I didn't want to see a story about two guys (one lovable loser and one lovable scoundrel) coming to terms with finally, after a prolonged adolesence, growing up.
I'm sure my impression of the movie is wrong. I haven't read any reviews, but the review headlines seem to say good things about it. I may yet go and see it. Or wait for DVD. :^)
P.S. Yuris, when are you going to start writing movie reviews for us?
I've been a long fan of the work of Wes Anderson and can't hardly wait for The Life Aquatic, coming out this Christmas.
Well, I'm always about a year behind in movies, but in the last week I've watched Eternal Sunshine of the Spotless Mind, Elf, and The Day After Tomorrow, and I recommend all of them - but for different reasons, of course.
Oh, and don't listen to the naysayers - The Day After Tomorrow is the best disaster epic since Deep Impact, if you know what I mean and I think you do.
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