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Topic: expected number of Runs (Read 4056 times) |
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inexorable
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expected number of Runs
« on: Jun 14th, 2011, 1:39pm » |
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There are 26 black(B) and 26 red(R) cards in a standard deck. A run is a maximum block of consecutive cards of the same color. For example, a sequence RRRRBBBRBRB of only 11 cards has
6 runs; namely, RRRR, BBB, R, B, R, B.
Find the expected number of runs in a shuffled deck of cards.
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ThudnBlunder
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Re: expected number of Runs
« Reply #1 on: Jun 14th, 2011, 2:08pm » |
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Voila
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inexorable
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Re: expected number of Runs
« Reply #2 on: Jun 15th, 2011, 10:19am » |
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I dint understand the last step 51*(26/51) 
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ThudnBlunder
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Re: expected number of Runs
« Reply #3 on: Jun 15th, 2011, 3:56pm » |
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on Jun 15th, 2011, 10:19am, inexorable wrote:| I dint understand the last step 51*(26/51) |
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inexecrable, each of these 51 other cards contributes to the expected numnber of runs and each contributes equally.
How much a card contributes depends on the probability that the next card is different.
If this probability were 0 (ie. cards are all the same) then the formula gives (51*0) + 1 = 1 run of length 52
If this probability were 1 (ie. cards alternate in colour) then the formula gives (51*1) + 1 = 52 runs, each of length 1
However, the actual probability = 26/51 = Number of different cards/Total number of other cards
So we get 51(26/51) + 1 = 27
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| « Last Edit: Jun 16th, 2011, 12:30am by ThudnBlunder » |
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ThudnBlunder
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Re: expected number of Runs
« Reply #4 on: Jul 5th, 2011, 10:07pm » |
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on Jun 14th, 2011, 2:08pm, ThudnBlunder wrote:
By the way, I no longer recommend the above linked site for anything, as it practises intellectual censorship. To explain, a couple of weeks ago I made a comment there in a thread about whether Maths is discovered or invented. One poster mentioned the Kronecker quote: "God created the natural numbers, and all else is the work of man." I replied that if God created all the natural numbers then how come he was able to take Sunday off, and adding as an afterthought in a further comment that he probably spent Sunday inventing the empty set for us to discover. These comments were immediately labelled as "unhelpful" and "out of place" by a self-styled mathematician and regular poster. I patiently explained that I was attempting to make the following points in a light-hearted manner (while not forgetting to append an 'imho'):
a) God has no need of duality.
b) The empty set does not exist.
I then suggested that knee-jerk reactions from 1D perspectives ought to be out-of-place, but seemingly were not. Yo, the next time I logged on the whole conversation had vanished into the ether. (Luckily for them, in cyberspace nobody can hear you scream.) Hence "Voila" doesn't exactly cut it for a site that applies such dishonest and cavalier moderation and I hereby retract any mention of it.
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| « Last Edit: Jul 10th, 2011, 11:54am by ThudnBlunder » |
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