A greedy Scrabble player (like the one most electronic implementations compare you to) always plays the highest scoring word available. This is the distribution of scores such a player earns on his first turn (using the North American English tile set and the SOWPODS dictionary). Note that a bar does not represent the probability of being able to achieve a given score, but rather the probability that a given score is the best that can be achieved. A cumulative distribution is also available, giving the probability that at least a given score can be achieved.
Only even scores are possible on the first move. Due to the comparatively high value of a bingo (50 points), the distribution is bimodal.
Famously, the highest possible opening play is MUZJIKS for 128 points, though the odds of being able to play it are less than 1 in 50,000,000. You're far more likely to find you have no scoring moves at all (0.6% of draws). The median draw can score just 28 points.
There are 16,007,560,800 possible ways to draw 7 tiles, resulting in 3,199,724 unique racks. These were searched exhaustively to produce this dataset.
This turned out to be a little computationally intensive for Python but with some clever optimizations, it only took about 4 hours to exhaust. Of course those clever optimizations probably took longer to come up with than just porting to C would have...