A question on probability please help (question in details)?

There are two possible sequences with alternating colors, depending on which color appears first.

Each marble that appears, of course, reduces the number of that color in the bag.

So if the first marble is green, so the alternating sequence is G B G B G B,

the probability of that sequence appearing is

(7/12) (5/11) (6/10) (4/9) (5/8) (3/7)

= (7 * 6 * 5 * 5 * 4 * 3) / (12 * 11 * 10 * 9 * 8 * 7)

= 12,600 / 665,280

If the first marble is blue, and the alternating sequence is B G B G B G,

the probability of that same sequence appearing is

(5/12) (7/11) (4/10) (6/9) (3/8) (5/7)

= (7 * 6 * 5 * 5 * 4 * 3) / (12 * 11 * 10 * 9 * 8 * 7)

which is exactly the same as for the green-first sequence.

[If you look carefully at the patterns of numerators and denominators in the first line of each calculation,

you will probably see why. They're the same, except for each pair of numerators being swapped.]

Since the overall probability of one or the other alternating sequence is just the sum of these,

we can just add a factor of 2 to one numerator in the factored single-fraction version,

which will make it easier to reduce to lowest terms:

(7 * 6 * 5 * 5 * 4 * 3 * 2) / (12 * 11 * 10 * 9 * 8 * 7)

= 6 * 5 / (11 * 9 * 8)

= 5 / (11 * 3 * 4)

= 5/132

= about 3.8%