What is the initial probability matrix? What is the market share for each size in the next round if purchase?

1. The final line of the problem states the original market shares for 1 and 2 liter containers. The matrices in the referenced picture have the columns labeled 1L and 2L with the row labeled P_0_, which is the initial probability distribution. Only two of the four choices have the correct numbers (0.55 and 0.45). Matching the column heading with the appropriate probability value will give the correct solution - the second choice.

2. Since 30% of 1L buyers switch to 2L containers, 70% stay with the 1L container. Similarly, since 15% of 2L buyers switch, 85% do not switch. Since the initial matrix has 1L containers in the first column and 2L containers in the second, the transition matrix should reflect this.

The transition matrix should reflect how the percentage of buyers of the two types of containers changes after a round of purchases. Matrix multiplication is row x column. In both matrices the first column represents 1L containers, the second column represents 2L containers. When multiplying row x column, it is first element of first row x first element of first column + second element of first row x second element of first column.

Again, the first column represents 1L containers. In the transition matrix the first row will represent 1L containers and the second row will represent 2L containers. 70% of 1L customers will stay 1L customers, while 15% of 2L customers will become 1L customers, so the first column of the transition matrix needs to be 0.7 and 0.15. Similarly, 30% of 1L customers switch to 2L, while 85% of 2L customers remain 2L customers. The second column needs to be .3 and .85. The third choice is correct.

3. Multiply the matrices as described earlier (admittedly a very brief description) and the resultant matrix is [0.4525, 0.5475].

4. The matrix found in part 3 replaces the original matrix and is multiplied by the transition matrix. The result is [0.3989, 0.6011].