A math teacher will push at this problem from the rear, so to speak - using a binomial calculator.
If a coin has a 53% chance of landing heads, then in 19 flips it has only a 5.5% chance of coming up heads 14+ times. That's the fact. Of course, the chance of it coming up 5 heads in a row (as the M's just did!) is only 4.2%. And we all know intuitively that it wasn't all that big a deal for the M's to win those 5 games.
But you're right: there is something about the nature of this game that makes the Mariners very unlike a flipped coin.
.........
Fangraphs pulls at the problem from the front of the golf cart: it reverse-engineers from game 162. It runs simulations of all games actually on the schedule, and factoring in Kyle Seager's 900 OPS+, etc.
This gives over a 15% chance for the M's to make the playoffs; undoubtedly your math prof amigo was not familiar with Fangraphs' simulated-seasons approach.
And! That 15% chance completely ignores any chance for the M's to suddenly be a better team now than they used to be.
.......
That 15% playoff chance must depend a lot on other teams losing, going 8-13, etc., more than on the M's going 14-5. No doubt, the M's chances depend on other teams opening the door. Their fate isn't in their own hands.
Great post as always Keith. Your point that 14-5 is very, very tough ... I hadn't realized quite how tough.