This time around, after you complete the first ride on your Multi Pass, you can reserve a fourth ride at any time slot after your first ride (rather than after your third ride). Similarly, after you have completed your second ride, you can reserve a fifth ride at any time slot after your second, and so on, until there are no available time slots remaining.
As before, the first three rides of the day are equally likely to be in any of the 12 time slots, whereas subsequent rides are equally likely to occur in any remaining available slots for the day.
On average, how many rides can you expect to “Lightning Lane” your way through today at Magic Kingdom?
As we say in the previous Classic answer, I encoded an "extra credit" version of the Lightning Pass program, where now as opposed to only adding new rides after your last scheduled ride, you add another ride uniformly distributed over all of the remaining, not already scheduled time slots. This distribution should be, and indeed turns out to be, much more centrally distributed, though here I was unable to come up with a semi-analytical solution and will just rely on the simulations.
Here again, I used N=1,000,000 simulations and arrived at the following histogram, which gives a mean of m=6.809379, standard deviation of s=1.404456, which implies that the true mean 95% confidence interval is R∈[6.8066,6.8121]. Therefore, let's comfortably round so that under the Extra Credit Lightning Lane rules, you can expect approximately 6.81 rides.
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