Well, from a purely mathematical "random" selection standpoint...the probability of winning the main game is:

With 6 black tiles: 3.6%

With 5 black tiles: 10.7%

With 4 black tiles: 21.4%

With 3 black tiles: 35.7%

With 2 black tiles: 53.6%

Now, the probability of getting a specific number of picks is more complicated, but basically

P(X=x) = nCx * p^x * q^(n-x)

Where p = Probability of success, q = probability of failure, n = # of tries, and x = number of correct answers.

In this case, p = 1/3, q = 2/3, n = 4, and x changes.

Getting rid of 0 tiles: 19.8%

Getting rid of 1 tile: 39.5%

Getting rid of 2 tiles: 29.6%

Getting rid of 3 tiles: 9.9%

Getting rid of 4 tiles: 1.2%

Therefore, the probability of:

Getting rid of 0 tiles and winning: 0.7%

Getting rid of 1 tile and winning: 4.2%

Getting rid of 2 tiles and winning: 6.3%

Getting rid of 3 tiles and winning: 3.5%

Getting rid of 4 tiles and winning: 0.6%

Therefore, the overall probability of winning is 15.3%

To put that into perspective, the probability of winning Gas Money from random selection is 20% I'm not fully sure how to calculate Pathfinders probability, so I won't do that. However, I do remember looking at probabilities a while back and finding most car games like Let 'em Roll, Any Number, and such hover around 25% and most in general hover between 20-30%. Even the probability of winning a game like Danger Price is 25%

However, most 2 prizers and quickies are between 25 - 50%, which is why 15.3% is very very hard for a 2 prizer. (And yeah, even though it's not a "quickie" I'm considering it to be one since all the other 2 prizers are)