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Here are five prominent or traditional bowls whose chances I'd rate as slim to none.
Sun Bowl: The Sun Bowl is the oldest non-BCS bowl and a former member of the Bowl Coalition—the 1992-94 forerunner of the BCS.
But it hasn't had marquee matchups (traditionally featuring third-place finishers or lower; or champions of mid-major), and is played in a small and antiquated stadium in a small media market.
Gator Bowl: After the Sun Bowl, the Gator Bowl is the second-oldest non-BCS bowl. It was a member of the Bowl Coalition, and later featured the runners-up of of the ACC and Big East Conferences.
I see it being hindered by not having been a particularly rich bowl in recent years, and for featuring a 6-6 team in it last year.
Russell Athletic Bowl: The Russell Athletic Bowl (formerly the ChampsSports Bowl) originated as the Blockbuster Bowl in 1990 on the premise of attracting big-name teams with a large payout.
It is currently one of five bowls to feature a BCS conference runner-up, albeit the runner-up of the unstable and noncompetitive Big East. It also suffers by not being one of the older or more prestigious bowls in the Southeast.
Holiday Bowl: The Holiday Bowl is the fourth-richest and third-oldest bowl west of the Rockies. However, it has fallen out of prestige since losing the Pac-12 runner-up, and is played in a very outdated Qualcomm Stadium.
Liberty Bowl: As with the Gator and Sun Bowls, the Liberty Bowl is another bowl that is more than half a century old (and is approaching half a century in Memphis). It currently features the champion of the mid-major Conference USA, and has also hosted the MWC and WAC champions in the past.
If the USA/MWC conference agreement reliably lands their champ in the Final Four, I'd give the Liberty Bowl a shot. Otherwise, no way.