Joe Biden - Winner - Being re-elected to the Vice-Presidency, Biden's odds of becoming president actually went down from when he was elected four years ago (29.55% to 25%). This is because his path to the presidency has changed. People elected president usually ascend to the White House following the President's death. People re-elected VP have always been elected on their own (and the number would be a good bit higher but for Gore's narrow loss). Still, had he lost re-election he would have joined the likes of Walter Mondale and Dan Quayle as Vice-Presidents who sought re-election and lost. Then his odds drop to 0%, so that's a pretty big change in odds.
Obama's new cabinet members - Winners - There will be new cabinet members, and some of them will find themselves elevated to a 0.38% chance of becoming president (For Senators, this is no improvement).
Romney's would-have-been cabinet members - Losers - Some of them would have been given Cabinet positions and for many of them, that would have increased their odds of becoming president (if only marginally), just as Obama cabinet members will.
The Romney sons - Losers - All are male and all are 35 years old or older but Craig. The four older sons would have seen their chances for becoming President go up to 3.23% and Craig's would be at 2.27% until he became old enough (none share Mitt's first name, but that doesn't matter now). Now their chances are pretty much the same as anyone else's - pretty close to 0%.
Paul Ryan - Loser - Being nominated was a big win for Ryan. It elevated him from the minuscule level of a House member (0.83%) to the level of a major party Veep nominee (13.33%) but had he actually won the Vice-Presidency, his chances would have jumped again to 29.55% instead of dropping to 7.45%. This somewhat overstates his chances. Most of those failed VPs who became Presidents were in the days when the Electoral College vote was a little less rigid. Since 1840, being the losing VP and going on to be elected President has only been pulled off once - by FDR. That would give him about a 1.5% chance, still nearly double where he was before.
Mitt Romney - Loser - Had he been elected, obviously his chances of becoming President would have gone to 100% (or perhaps less barring a sudden death or Electoral College scandal), but since he lost, his chances drop to 8.89% (though, in reality probably even lower).
If there are faithless electors, then there will be more chances for winners and losers, but I doubt that getting a faithless elector vote for President actually gives you a 1 in 5 chance of becoming President. It's probably more a recognition of your credibility - it doesn't make you more likely.
Below is the full table of rolls and probabilities.
|If you…||% chance||# of people who've done it||# who eventually became president|
|Win the popular vote for President†||93.75%||32||30|
|Come in first in the Electoral College||97.37%||38||37|
|Are appointed Vice-President||50.00%||2||1|
|Are elected Vice-President||29.55%||44||13|
|Are re-elected Vice-President||25.00%||12||3|
|Receive Electoral College votes for Vice-President||12.90%||124||16|
|- same as above less faithless electors||13.33%||120||16|
|- same as top and lose††||7.45%||94||7|
|- same as above less faithless electors||7.87%||89||7|
|Come in 2nd in the Electoral College*||12.50%||48||6|
|- same as above post 12th Amendment||8.89%||45||4|
|Are appointed Secretary of State**||9.23%||65||6|
|Come in 3rd or lower in Electoral College||4.44%||45||2|
|- same as above less faithless electors||2.50%||40||1|
|- only faithless electors||20.00%||5||1|
|- same as above post 12th Amendment||0.00%||19||0|
|Are elected Speaker of the House||1.89%||53||1|
|Are the legitimate child of a President||1.31%||153||2|
|- who lives to 35||2.35%||85||2|
|- are a son||2.27%||88||2|
|- who lives to 35||3.23%||62||2|
|- who shares his father's first name||12.50%||16||2|
|Become a Senator**||0.85%||1877||16|
|Are appointed a Cabinet official (other than State)**||0.38%||524||2|
|Elected to House***||0.18%||10251||18|
|Run for re-election as Vice-President and lose ‡||0.00%||3||0|
|†After record keeping began in 1824. Not all states are accounted for though. The Constitution specifies that the president and vice president be chosen through the votes cast by electors chosen by the states, rather than by a direct popular vote. At first, some electors were chosen by state legislatures, but by 1836 all states but South Carolina chose electors through a statewide popular vote. (S.C. followed suit in 1860.)|
|*John Qunicy Adams shows up twice because he came in 2nd then became president and then came in 2nd again|
|**Not counting those ineligible for Presidency|
|†† In place of nominated|
|‡ James Sherman was running for re-election for Vice-President in 1912, but died days before the election. His ticket came in 3rd. He is not counted here since he was ineligible for re-election.|
|*** Best number I could find. It's difficult to find an accurate list of all former house members, and thus to eliminate ineligible house members|