The statement of the problem I am facing is:

Arthur and Bernard participated of an election to take care of a building with 100 voters. At the end, 60 people voted in Arthur and 40 in Bernard. What is the chance that during the investigation of the result of the votes, Arthur is always ahead of Bernard?

However, I didn't find a direct way, even though, my idea was to find all sequences, such that Arthur gets 60 votes and the remaining is given to Bernard, as in A-, AAB, ABA, AABA, AAAB, this is, the number of sequences such that #A's > # B's and #A's + #Bs < 101.

  • 1
    $\begingroup$ My idea was to draw a grid with A votes on the x axis and B votes on the y axis. If we can count n=the number of routes from (0,0) to (60,40) where A is always greater than B we will be nearly done - because we can easily calculate m = the total number of routes (100 choose 40) and the required probability is n/m. It's easy to label the number of routes on the grid with the number of routes, but there are too many to do by hand and I can't find (yet) a useful closed form. $\endgroup$
    – Blitzer
    Sep 14, 2022 at 16:15
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    $\begingroup$ See Bertrand's Ballot Theorem $\endgroup$
    – lulu
    Sep 14, 2022 at 16:20
  • $\begingroup$ @lulu great! It would be better I guess for you to post that link as an answer while adding that formula as summary. Then the question could be marked as solved perhaps. $\endgroup$ Sep 14, 2022 at 16:29
  • $\begingroup$ @InanimateBeing not a fan of link only solutions. Feel free to post it yourself, under Community if you prefer. Or the OP could. $\endgroup$
    – lulu
    Sep 14, 2022 at 17:52
  • $\begingroup$ @lulu, in general I see you tend to answer more via comments than actually posting an answer. Example: quite some of my own doubts/posts that were solved by you but in comments. I find it strange and near-inexplicable but your style is still fine (maybe not wrt EoQS?). $\endgroup$ Sep 15, 2022 at 6:57


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