How to update a matrix of probabilities

I am trying to find/figure out a function that can update probabilities.

Suppose there are three players and each of them get a fruit out of a basket: ["apple", "orange", "banana"]

I store the probabilities of each player having each fruit in a matrix (like this table):

	apple	orange	banana
Player 1	0.3333	0.3333	0.3333
Player 2	0.3333	0.3333	0.3333
Player 3	0.3333	0.3333	0.3333

The table can be interpreted as the belief of someone (S) who doesn't know who has what. Each row and column sums to 1.0 because each player has one of the fruits and each fruit is at one of the players.

I want to update these probabilities based on some knowledge that S gains. Example information:

Player 1 did X. We know that Player 1 does X with 80% probability if he has an apple. With 50% if he has an orange. With 10% if he has a banana.

This can be written more concisely as [0.8, 0.5, 0.1] and let us call it reach_probability.

---

A fairly easy to comprehend example is:

probabilities = [
    [0.5, 0.5, 0.0],
    [0.0, 0.5, 0.5],
    [0.5, 0.0, 0.5],
]

# Player 1's 
reach_probability = [1.0, 0.0, 1.0]

new_probabilities = [
    [1.0, 0.0, 0.0],
    [0.0, 1.0, 0.0],
    [0.0, 0.0, 1.0],
]

The above example can be fairly easily thought through.

---

another example:

probabilities = [
    [0.25, 0.25, 0.50],
    [0.25, 0.50, 0.25],
    [0.50, 0.25, 0.25],
]

# Player 1's 
reach_probability = [1.0, 0.5, 0.5]

new_probabilities = [
    [0.4, 0.2, 0.4],
    [0.2, 0.5, 0.3],
    [0.4, 0.3, 0.3],
]

---

In my use case using a simulation is not an option. My probabilities matrix is big. Not sure if the only way to calculate this is using an iterative algorithm or if there is a better way.

I looked at bayesian stuff and not sure how to apply it in this case. Updating it row by row then spreading out the difference proportionally to the previous probabilities seems promising but I haven't managed to make it work correctly. Maybe it isn't even possible like that.

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