Consider a 10 mile length of road with mile markers exactly every mile including 0 and 10. Starting at the 0 marker, you drive towards 10 and randomly stop at some point with a position "p" defined as (0 marker)<= p <=(10 marker).

Part 1 (teaser): - What is the expected value of the distance from "p" to the nearest mile marker?

Part 2 (more difficult): - Three mile markers, randomly selected, have been removed by vandals, but 0 and 10 remain. Same question as part 1.

Part 3: - Consider part 2, but markers 0 and 10 are also potentially part of the vandalism.
Does the answer change from part 2, and if so what is it?
(You have no problem knowing where to start your trip, even if marker 0 happens to be missing)

Part 3

In agreement with prior posts I got
Part 1:  1/4 = 0.25
Part 2:  7/16 = 0.4375

My method for Part 3 is essentially what Charlie outlined but with one more case and different terminology

Part 3
Case       Name                Exp Val  #ways
1   Triple, no end                  0.55    7
2   Double & Single, no end   0.45   45
3   Singles, no end                0.4    35
4   End triple                        0.625   2
5   End double, mid single     0.45   14
6   End double, end single     0.425   2
7   Mid double, end single      0.425  14
8   All singles, one end          0.375  42
9   All singles, two ends         0.35    7

Multiply Exp Value * #ways for each case.
Sum those values and divide by the sum of the #ways.
The #ways is C(11,3) = 165 which agrees.

Final answer:  0.42
(edited to fix typo)

Edited on August 31, 2017, 6:34 pm

Posted by Larry on 2017-08-31 15:19:25