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An e-Xperiment!

Point your pen randomly at one of these digits of e...and hold it there...

2.71828182845904523536028747135266249775724709369995957496696762
7724076630353547594571382178525166427427466391932003059921817413
5662904357290033429526059563073813232862794349076323382988075319
5259101901157383418793070215408914993488416750924476146066808226
4800168477411853742345442437107539077744992069551702761838606261
3313845830007520449338265602976067371132007093287091274437470472
3069697720931014169283681902551510865746377211125238978442505695
3696770785449969967946864454905987931636889230098793127736178215
4249992295763514822082698951936680331825288693984964651058209392
3982948879332036250944311730123819706841614039701983767932068328
2376464804295311802328782509819455815301756717361332069811250996
1818815930416903515988885193458072738667385894228792284998920868
0582574927961048419844436346324496848756023362482704197862320900
2160990235304369941849146314093431738143640546253152096183690888
7070167683964243781405927145635490613031072085103837505101157477
0417189861068739696552126715468895703503540212340784981933432106
8170121005627880235193033224745015853904730419957777093503660416
9973297250886876966403555707162268447162560798826517871341951246
6520103059212366771943252786753985589448969709640975459185695638
0236370162112047742722836489613422516445078182442352948636372141
7402388934412479635743702637552944483379980161254922785092577825
6209262264832627793338656648162772516401910590049164499828931505...

Now, have someone roll a single die. Continuing to the right along the e-sequence, slowly slide your pen over that number of digits, being sure to record those digits as a decimal number between 0 and 1. Continue this process by rolling the die again, moving your pen, and recording the number string as a decimal. Add the two numbers. If the sum is greater than 1, repeat the sequence...etc. until the sum exceeds 1.

Example: Suppose you place your pen on the second 7 in the second row.
You roll a "6"...so you record the number 0.240766
You roll a "1"...so you record the number 0.3, add and get the sum 0.540766
You roll a "4"...so you record the number 0.0353, add and get the sum 0.576066
You roll a "3"...so you record the number 0.547, add and get the sum 1.123066
Stop the experiment.

The Question: If this experiment is repeated a great many times, what is the expected number of rolls necessary to to reach a sum greater than 1?

Note: If you have been following the theme of this week, you should be able to guess the answer!

Source: Variation of H. Shultz's problem TYCMJ, September 1979, pp. 277-278