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I am trying to simulate a process of selection without replacement. The process is one in which the system places a set of items in a specific order, and then the user selects N items in whatever order they want. The order of the items will vary depending on an algorithm.

The probability that each item will be picked by the user varies based on: the item, the position, and which draw the user is on (1st, 2nd, etc.).

The full probability space is: 1 = Pr(Draw any item) + Pr(Draw no items and end selection)

+--------+----------+
|  Item  | Position |
+--------+----------+
| Item A |        1 |
| Item B |        2 |
| Item C |        3 |
| Item D |        4 |
+--------+----------+

My challenge is trying to figure out how to generate a conditional probability table that can be used in a simulation. I have one that is Pr(Click | Item, Position, Draw Number). But what I need is Pr(Click | Item, Position, Draw Number, {History of draws})

+--------+----------+------+-------------+
|  Item  | Position | Draw | Probability |
+--------+----------+------+-------------+
| Item A |        1 |    1 | 0.33        |
| Item B |        2 |    1 | 0.25        |
| Item C |        3 |    1 | 0.2         |
| Item D |        4 |    1 | 0.14        |
| Item A |        1 |    2 | 0.1         |
| Item B |        2 |    2 | 0.3         |
| Item C |        3 |    2 | 0.18        |
| Item D |        4 |    2 | 0.05        |
+--------+----------+------+-------------+

This table fails to take into account that the process is a draw without replacement. The probabilities for draw 2 do not take into account which item was selected in the first draw.

If I simulate a series of draws, and simulate the user on Draw 1 selecting Item B, is there a way to modify the probabilities of Items A, C D for draw 2 to account for the fact that Item B is no longer eligible?

I think there is an application of Bayes Theorem here, but I'm struggling with how to adapt bayes formula to account for the number of conditionals.

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