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A personal lotto game (Posted on 2009-06-12) Difficulty: 3 of 5
32 members of our social group use the weekly results of our national lotto draw for our competition. Each person selects a set of 3 numbers from just one of the coloured grids of the map shown below, such that they are all in a row (e.g. 31, 32 and 33) or in a column (e.g. 23, 29 and 35) or diagonally (e.g. 06, 11 and 16):


01 02 03 04 05 06
07 08 09 10 11 12
13 14 15 16 17 18
19 20 21 22 23 24
25 26 27 28 29 30
31 32 33 34 35 36



Our national lotto draws, each week, 5 numbers from the set of 36 numbers, and we permanently mark these 5 numbers in our map. The winner in our competition will be the person who firstly has his 3 numbers marked.

What is the probability that our competition:
a) ends in the first week?
b) ends in the second week?
c) ends in the third week?

And the most important:
d) What is the expected number of weeks for us to have a winner?

See The Solution Submitted by pcbouhid    
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Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts simulation | Comment 2 of 4 |

After 1,000,000 trials of a simulation, the following are the number of times that it took the given number of weeks:

weeks:         1       2       3       4      5        6       7       8  9  more
occurrences: 44471  324823  402107  182870  39741    5341     597     50  0  0
percent:    4.4471 32.4823 40.2107 18.2870 3.9741  0.5341  0.0597 0.0050  0  0

The mean number of weeks (serving as an estimate of the expectation) is 2.867248.

FOR trial = 1 TO 1000000
   REDIM bd(4, 3, 3)
   hit = 0
   FOR drwng = 1 TO 9
     REDIM used(36)
     FOR chc = 1 TO 5
      DO
       r = INT(RND(1) * 36)
      LOOP WHILE used(r)
      used(r) = 1
      quad = INT(r / 9) + 1: r = r MOD 9
      row = INT(r / 3) + 1
      col = (r MOD 3) + 1
      bd(quad, row, col) = 1
     NEXT
     FOR quad = 1 TO 4
       FOR row = 1 TO 3
         f = 1
         FOR col = 1 TO 3
           IF bd(quad, row, col) = 0 THEN f = 0: EXIT FOR
         NEXT
         IF f THEN hit = 1: EXIT FOR
       NEXT
       IF hit THEN EXIT FOR
       FOR col = 1 TO 3
         f = 1
         FOR row = 1 TO 3
           IF bd(quad, row, col) = 0 THEN f = 0: EXIT FOR
         NEXT
         IF f THEN hit = 1: EXIT FOR
       NEXT
       IF hit THEN EXIT FOR
       f = 1
       FOR row = 1 TO 3
         col = row
         IF bd(quad, row, col) = 0 THEN f = 0: EXIT FOR
       NEXT
       IF f THEN hit = 1: EXIT FOR
       f = 1
       FOR row = 1 TO 3
         col = 4 - row
         IF bd(quad, row, col) = 0 THEN f = 0: EXIT FOR
       NEXT
       IF f THEN hit = 1: EXIT FOR
     NEXT
     IF hit THEN EXIT FOR
   NEXT
   stat(drwng) = stat(drwng) + 1
   IF trial MOD 10000 = 0 THEN
    FOR i = 1 TO 10: PRINT stat(i); : NEXT: PRINT trial
   END IF
NEXT trial
PRINT
expect = 0
FOR i = 1 TO 10
  expect = i * stat(i) / (trial - 1) + expect
NEXT
PRINT expect


 


  Posted by Charlie on 2009-06-12 15:57:05
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