Marbles in a bag: what's the chance of a specific combination with multiple picks

Question

1000 marbles are in a bag, each with a letter letter A - K

 

A: 109 Marbles

B: 109 Marbles

C: 109 Marbles

D: 109 Marbles

E: 109 Marbles

F: 109 Marbles

G: 109 Marbles

H: 109 Marbles

I: 109 Marbles

J: 14 Marbles

K: 5 Marbles

 

Each time a marble is picked, the marble is put back in in the bag for the next pick. 

 

With 18 picks, what's the chance of picking at least one of each letter? Order does not matter.

 

 

Answer 1

For illustration let’s first use a bag with three balls A, A, B, and three draws. The probability of picking A or B, P(A or B) in three draws is 1 minus the overhang , P(AAA) or P(BBB). P(A or B) – [P(AAA) + P(BBB)] in three draws. Addition is used for non-mutually exclusive events and multiplication for mutually exclusive events. For three draws this equals 1 – [(1/3)*(1/3)*(1/3] – [(2/3)*(2/3)*(2/3] = 1 – 9/27 = 18/27 that can be verified by hand construction. For this problem the joint probability of all A= P(A) for one draw is 109/1000. For eighteen draws it is 18*109/1000 and is the same for C, B, D, E, F, G, I, J, K. For J, 18*14/1000, K,18*5/1000. Using mutual and non-mutual events as in the initial problem we have the probability of at least one letter in eighteen picks is
1 – [9*(109/1000)^18 + (14/1000)^18 +( 5/1000)^18]~1 in 16 digit arithmetic. If one could carry more accurate arithmetic an “exact” answer could be found.