I already have a post this season called Lovie's Blunder, but I guess I'm just not terribly imaginative, much like many coaches in the NFL.
In a tight game against the Redskins, Smith decided not to challenge Jay Cutler’s fumble on first and goal from the one. Most accounts claim the replay clearly showed the ball crossed the plane of the end zone prior to the fumble.
Had the play been a touchdown, the Bears would have been up 11 points early in the third quarter, giving them only an 89% chance of winning. Instead, the fumble put the Bears’ chances at 70%, a difference of 19%. To put that in perspective, only two of the game's plays were more significant, and they were the two interception returns for touchdowns.
Smith had lost a challenge on the prior play, and you have to think that must have affected his judgment.
Just for the sake of argument, let's assume that the Bears eventually found themselves down by 3 points inside the two-minute warning. (They were up by 4 at the time of the play.) If we value a timeout as a full 40 seconds between plays, the difference in Win Probability (WP) between having 2:00 left to play and 1:20 left to play at their 30 yard line would have been 0.04 WP (0.24 WP compared to 0.20 WP). The difference between having 60 and 20 seconds to play is 0.07 WP. The biggest difference in the value of a timeout is between 40 seconds and zero seconds (a certain loss), which is 0.10 WP. Assuming the challenge had a reasonable probability of being upheld, he should have thrown the flag.
One of the cool things about a WP model is that it's a linear utility function, so we can evaluate gambles like the decision to challenge or not. For now let's set aside the value of consuming his second and final coach's challenge. Assuming the largest value of the timeout (0.10 WP), the break-even probability that the challenge would be upheld (x) would need to be:
0.19 x = 0.10 (1-x)
x = 0.34
So Smith would only need a 34% chance of being upheld for the challenge to be worth it, even assuming the worst-case scenario that the Bears would be down by a single score at the end of the game. You don't need algebra to figure out that play should have been challenged. There aren't going to be many reviewable plays all year as big as that--it's either a TD or a turnover.