Alright, let's calculate the probability of Alan picking a white marble on his 5th try. To do that, we need to consider the number of white marbles left in the bag and the total number of marbles left after each turn. Let's break it down step by step:
On the first try, Alan drops 1 black, 2 green, and 1 white marble. So, the total number of marbles left in the bag is 5 + 8 + 4 + 2 - (1 + 2 + 1) = 15.
On the second try, Alan drops another 1 black, 2 green, and 1 white marble. So, the total number of marbles left in the bag is 15 - (1 + 2 + 1) = 11.
On the third try, Alan drops 1 black, 2 green, and 1 white marble again. So, the total number of marbles left in the bag is 11 - (1 + 2 + 1) = 7.
On the fourth try, Alan drops 1 black, 2 green, and 1 white marble once more. So, the total number of marbles left in the bag is 7 - (1 + 2 + 1) = 3.
Finally, on the fifth try, Alan picks a marble. Since there are 3 white marbles left in the bag and the total number of marbles is 3, the probability of Alan picking a white marble on his 5th try is 3/3, which is equal to 1.
So, the probability of Alan picking a white marble for the first time on his 5th try is 1.