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The flip side is that the number of outcomes that falls within the set of easily recognizable patterns is extremely small: probably not too much larger than the following ...
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What about:
1,1,2,2,1,1,2,2,1,1
for example (or any sequence like it) -- these are possibly generated by the process in question too.
I had read others' posts before writing mine and many of them confused the process by NOT replacing items once selected. I inadvertently carried that method over into my thought processes.
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Therefore, while we should not be surprised to hear that "Frank" hit upon your sequence by chance in a single shot, we should be surprised to hear that "Frank" hit upon the sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 by chance in a single shot (in fact, we should probably reject "Frank"'s claim).
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Pick a sequence in advance (any sequence) and I would be suprised to see it first try no matter what.
The odds of getting one pre-specified result is 1 in 10 billion,
the odds of getting the result you just got is 1 in 1.
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First, odds and probability are not the same thing. If the probability of event E occurring is P(E) = 0.25 (in other words, 1 in 4 chance), the odds for E are 1:3 (and the odds against E are 3:1).
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1) The probability of getting one particular pre-specified result is 1 in 10 billion.
2) The probability of getting one particular unspecified result is 1 in 10 billion.
The act of specifying here does not influence the future outcomes, in that, as others stated, the probability of any single outcome is the same as any other: selecting one of them beforehand does not change its probability of occurring - it's probability 1 in 10 billion whether specified or not.
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1) The probability that any event that did occur did occur is 1 (absolute certainty), no matter how unlikely it was to occur prior to its occurring.
This fact can be misused/misapplied. If I roll 5 dice together one time and get a Yahtzee, is the probability of getting a Yahtzee in a single try 1? No. The probability that I got a Yahtzee in a single try
given that I got a Yahtzee in a single try is 1: that's after the fact. It does not change the fact that the probability of getting a Yahtzee in a single try is 1 in 6^4, and that that probability was the case prior to the attempt.
[This message has been edited by DNAunion, 11-27-2003]