Normalizing logistic regression probabilities to fixed sum

#1
I have run a logistic regression model on a target variable and get a list of probabilities like [0.50, 0.30, 0.20, 0.10, 0.05, 0.05, 0.01].

For the target situation, I know that there are always going to be 3 positive results. I’m looking at a soccer league and comparing some stats (goals, last year rank, etc) to predict likelihood of top 3 in a given year. Each previous subgroup/year will have 3 positive results. However, since the probabilities calculated are independent of each other, the predicted values for next year’s group will not sum to 3. Since I know the group of teams for this year, I would want the sum of the predicted values to equal my target of 3.

I'm wondering if there is a way to properly scale/adjust these probabilities so that they add up to 3, obviously without any going over 1 (as they would if I simply scaled by sum). I am currently using scaled up/down probabilities (summing to 1) based on their proportion to the group and then simulating results based on these probabilities, but I'm not sure if this is mathematically sound.

I'm open to other approaches and would love some guidance. Thanks!
 
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