My reading of this question is calculate p(n) the probability that exactly n rolls are needed to have at least one of each possible result 1,2,3,4,5,6.
You can start by assuming that the last roll to achieve a complete set is a 1. Then for the n-1 earlier rolls we have a binomial experiment where we want to know the probability of n-1 successes at a probability of 1/(n-1), and for the n-th roll we have a probability of success (getting a 1) of 1/6.
CB
Wrote short simulation program; 5 runs @ 10 million tries each:
147,011,358
147,033,575
147,000,132
146,994,434
147,035,012
So looks like slightly over 14.7 rolls
If wrong, can anyone tell me what's wrong with my program:
100 set up array A(6)
110 for g = 1 to 10000000
120 array A = 0
130 r = 1 'r = rolls
140 k = random number 1 to 6
150 A(k) = 1
160 if array A = 6 then goto 180
170 r = r + 1:goto 140
180 t = t + r
190 next g
200 print t
This is the "Coupon Collector's Problem". See
Coupon collector's problem - Wikipedia, the free encyclopedia
Thanks CB, you're telling me I misunderstood; I took the problem to mean:
after how many rolls will each of 1 to 6 appear at least once; example:
3,5,3,1,2,6,1,5,6,6,2,1,3,5,4: 15 rolls
After a number (1 to 6) is picked, program "flags" the corresponding array
number with "1"; like, after picking the 3 at start, program makes A(3) = 1.
When the last pick (4) was made, program made A(4)=1:
at this point, all array locations were equal to 1, so array totalled 6.
So this session was completed, and program goes to line 180 in order
to update the total rolls so far.
Can you tell me what I'm missing? Thanks.
Nothing, if you understand what that line means.
Though I will say this: the syntax of the language is unacceptable in that it makes no distinction between the assignment operator and the logical equality operator, and the "array" qualifier seems to be overused and the function notation is counter-intuitive and non-standard. Also profligate use of GOTOs in modern code is deprecated.
CB
If you are using pseudo-code you should get rid of line numbers, it should be readable without being confused with real code:
Code:set Nreplications=10000 //number of replications of the experiment set array A[1..6] to zero vector //array of flags to indicate if a face has come up in an experiment set counter=0 //counts number of throws in an experiment for idx=1 to Nreplications for throw=1 to infinity set A to zero vector set k to UniformRand on {1,2,..6} set A[k] to 1 if (sum of A[k], k=1,..6) is equal to 6 set total to total+throw break out of throw loop end if end for end for output total/Nreplications