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.

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