Xing glass bridges [or not]

This article is originally published at https://xianblog.wordpress.com

A riddle from the Riddler surfing on Squid Games. Evaluating the number of survivors (out of 16 players) able to X the glass bridge, when said bridge is made of 18 consecutive steps, each involving a choice between a tempered and a non-tempered glass square. Stepping on a non-tempered square means death, while all following players are aware of the paths of the earlier ones. Each player thus moves at least one step further than the previous and unlucky player. The total number of steps used by the players is therefore a Negative Binomial Neg(16,½) variate truncated at 19 (if counting attempts rather than failures), with the probability of reaching 19 being .999. When counting the number of survivors, a direct simulation gives an estimate very close to 7:

   mean(apply(apply(matrix(rgeom(16*1e6,.5)+1,nc=16),1,cumsum)>18,2,sum))

but the expectation is not exactly 7! Indeed, this value is a sum of probabilities that the cumulated sums of Geometric variates are larger than 18, which has no closed form as far as I can see

   sum(1-pnbinom(size=1:16,q=17:2,prob=.5)

but whose value is 7.000076. In the Korean TV series, there are only three survivors, which would have had a .048 probability of occurring. (Not accounting for the fact that one player was temporarily able to figure out which square was right and that two players fell through at the same time.)

Looking later at on-line discussions, I found that the question was quite popular, with a whole spectrum of answers… Including a wrong Binomial B(18, ½) modelling that does not account for the fact that all 16 (incredibly unlucky) players could have died before the last steps.

And reading the solution on The Riddler a week later, I was sorry to see this representation of the distribution of survivors, as if it was a continuous distribution!

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This article is originally published at https://xianblog.wordpress.com
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