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        riddles >> medium >> expected number of Runs
        
(Message started by: inexorable on Jun 14^th, 2011, 1:39pm)

Title: expected number of Runs
Post by inexorable on Jun 14^th, 2011, 1:39pm

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.

Title: Re: expected number of Runs
Post by ThudnBlunder on Jun 14^th, 2011, 2:08pm

Voila (http://math.stackexchange.com/questions/2763/what-is-the-expected-number-of-runs-of-same-color-in-a-standard-deck-of-cards)

Title: Re: expected number of Runs
Post by inexorable on Jun 15^th, 2011, 10:19am

I dint understand the last step 51*(26/51) :(

Title: Re: expected number of Runs
Post by ThudnBlunder on Jun 15^th, 2011, 3:56pm

on 06/15/11 at 10:19:17, inexorable wrote:

I dint understand the last step 51*(26/51) :(

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

Title: Re: expected number of Runs
Post by ThudnBlunder on Jul 5^th, 2011, 10:07pm

on 06/14/11 at 14:08:05, ThudnBlunder wrote:

Voila (http://math.stackexchange.com/questions/2763/what-is-the-expected-number-of-runs-of-same-color-in-a-standard-deck-of-cards)

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.