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The Last Diamond (Posted on 2004-05-25) Difficulty: 3 of 5
A standard, thoroughly shuffled 52-card deck is dealt one at a time to 5 players (players 1 - 5) in standard fashion, until the deck is exhausted.

Using non-brute force methods, show which player is most likely to be dealt the last Diamond in the deck.

See The Solution Submitted by Brian Smith    
Rating: 2.8000 (5 votes)

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The actual numbers (spoiler present) | Comment 2 of 5 |

The calculated probability of getting the last diamond on a particular card is specified in the table below by the particular card dealt, and then is summarized in overall probabilities for players 1 through 5:

0.00000000   0.00000000   0.00000000   0.00000000   0.00000000
0.00000000   0.00000000   0.00000000   0.00000000   0.00000000
0.00000000   0.00000000   0.00000000   0.00000000   0.00000000
0.00000000   0.00000000   0.00000001   0.00000003   0.00000008
0.00000020   0.00000046   0.00000102   0.00000213   0.00000426
0.00000819   0.00001521   0.00002738   0.00004791   0.00008172
0.00013621   0.00022223   0.00035557   0.00055876   0.00086353
0.00131407   0.00197110   0.00291723   0.00426365   0.00615860
0.00879801   0.01243856   0.01741399   0.02415488   0.03321296
0.04529041   0.06127525   0.08228391   0.10971188   0.14529412
0.19117647   0.25000000
last diamond
0.24672355   0.32592283   0.10299911   0.13873924   0.18561528

and was obtained by:

DEFDBL A-Z
CLS
prNotThisCum = 1
FOR card = 52 TO 1 STEP -1
  pl = (card - 1) MOD 5 + 1
  round = (card - 1) \ 5 + 1
  prNotThis = (card - 13) / card
  prThis = prNotThisCum * 13 / card
  prNotThisCum = prNotThisCum * prNotThis
  tProb(pl) = tProb(pl) + prThis
  LOCATE round + 1, (pl - 1) * 13 + 1
  PRINT USING "#.########"; prThis;
NEXT card

FOR pl = 1 TO 5
 LOCATE 15, (pl - 1) * 13 + 1
 PRINT USING "#.########"; tProb(pl);
NEXT

If the problem had called for, say, the last ace (only 4 in the deck rather than 13), the probabilities would have been:

0.00000000   0.00000000   0.00000000   0.00000369   0.00001478
0.00003694   0.00007388   0.00012928   0.00020685   0.00031028
0.00044325   0.00060947   0.00081263   0.00105642   0.00134454
0.00168067   0.00206852   0.00251177   0.00301413   0.00357928
0.00421092   0.00491273   0.00568843   0.00654169   0.00747622
0.00849571   0.00960384   0.01080432   0.01210084   0.01349709
0.01499677   0.01660356   0.01832117   0.02015329   0.02210361
0.02417582   0.02637363   0.02870071   0.03116077   0.03375750
0.03649460   0.03937575   0.04240465   0.04558500   0.04892049
0.05241481   0.05607166   0.05989473   0.06388771   0.06805430
0.07239819   0.07692308
last ace

0.21534768   0.23261612   0.16926771   0.18371041   0.19905808

 


  Posted by Charlie on 2004-05-25 08:59:04
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