Could you explain solution of this question step by step? Suppose that balls are

Question

Could you explain solution of this question step by step?

Suppose that balls are successively distributed among 8 urns, with each ball being equally likely to be put in any of these urns. What is the probability that there will be exactly 3 nonempty urns after 9 balls have been distributed?

Solution: If we let Xn be the number of nonempty urns after n balls have been distributed, then Xn, n 0 is a Markov chain with states 0, 1, . . . , 8 and
transition probabilities
Pi,i = i/8 = 1 Pi,i+1, i = 0, 1, . . . , 8
The desired probability is P90,3 =P81,3 where the equality follows because P0,1 =1. Now, starting with 1 occupied urn, if we had wanted to determine
the entire probability distribution of the number of occupied urns after 8 additional balls had been distributed we would need to consider the transition probability matrix with states 1, 2, . . . , 8. However, because we only require the probability, starting with a single occupied urn, that there are 3 occupied urns after an additional 8 balls have been distributed we can make use of the fact that the state of the Markov chain cannot decrease to collapse all states 4, 5, . . . , 8 into a single state 4 with the interpretation that the state is 4 whenever four or more of the urns are occupied. Consequently, we need only determine the eight-step transition probability P81,3 of the Markov chain with states 1, 2, 3, 4 having transition probability matrix P given by

Raising the preceding matrix to the power 4 yields the matrix P4 given by

Hence,
P81,3= 0.0002 × 0.2563 + 0.0256 × 0.0952 + 0.2563 × 0.0198+ 0.7178 × 0 = 0.00756

Explanation / Answer

We need to find the probability of exactly 3 occupied urns after 9 balls have been distributed. Here, we don’t choose 8 states because Markov chain cannot decrease and treats the 4th state as 4 or more urns occupied. if the chain is in state i at time n, then at time n + 1, it will either stay in state i with probability i/8 or it will transit to i + 1 with probability 1 – i/8 i.e ., Pi,i =i/8 = 1 – i/8 . Thus the transition probability if formed by

P =

              [,1] [,2] [,3] [,4]

   [1,] 0.125 0.875 0.000 0.000

   [2,] 0.000 0.250 0.750 0.000      #4th state represent 4 or more which obviously will

   [3,] 0.000 0.000 0.375 0.625        be P8,8 = 1.

   [4,] 0.000 0.000 0.000 1.000

We are supposed to find P1,38 ( from 1 occupied urn to 3 occupied in the next step). In order to find this, calculate P8 and the answer is (1,3) element.

                  [,1]                [,2]                 [,3]              [,4]

[1,] 5.960464e-08 1.063943e-04 0.0075727701 0.9923208

[2,] 0.000000e+00 1.525879e-05 0.0022548437 0.9977299

[3,] 0.000000e+00 0.000000e+00 0.0003910661 0.9996089

[4,] 0.000000e+00 0.000000e+00 0.0000000000 1.0000000

P1,38 = 0.00757

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