Linear Model to generate probability of each possible output
From what I understood - 

Train a classifier with the features as you mentioned and the output probabilities like you have shown - 


So, for Player A, you can use all the features (average runs scored overall, average score in past 10 innings etc.)
Treat the runs scored as a categorical variable and try to learn a classifier
Put it through the network with a softmax layer at the end - that will give you Sum of Probability should add up to 1


My recommendation


Each player will have scored different runs so you will not get enough data points in a particular category (so no. of people who score 83 will be really less for example)
Pre-group the ranges in your training data and then do the classifier. 
Having a softmax layer at then end won't give you the "linear model" you want, but would definitely solve the problem with a lot more clarity.