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.

Asking about the probability of getting the last diamond, is the same as asking about the probability of getting the first diamond if the deck was dealt from the bottom up.

Odds that the 1st card is the first diamond: 13/52

Odds that the 2nd is it: 39/52.13/51

Odds that the 3rd is it: 39/52.38/51.13/50

Odds that the 4th is it: 39/52.38/51.37/50.13/49

...and so on

Using OpenOffice Calc, the odds favor the 2nd player, who gets the last card: 32.6%. The 1st player is next: 24.7%. The 5th has 18.5%, the 4th 13.9%, and the 3rd, 10.3%.

If there were only 4 players, odds would have favored the 4th: 36.4%, 27.5%, 20.7% and 15.4%.