You only have four things to pick from, so you can calculate each step explicitly. This equation follows the notation in the article: T is the time to collect all of the items, and ti be the time to collect the i-th item after i−1 items have been collected.

E(T)=E(t1)+E(t2)+E(t3)+E(t4)=p1−1+p2−1+p3−1+p4−1.
The probability p1 of picking a new one if you have picked none yet is 1.

The probability p2 of picking a new one if you have picked one depends on which one you picked first. Let's call the items A through D and the probabilities of picking each item if they're all in the box pa through pd.

Then p2 would be
pa(pb+pc+pd)+pb(pc+pd+pa)+pc(pd+pa+pb)+pd(pa+pb+pc).
The above expression is the probability of picking a new one given that A had been picked already, plus the probability of picking a new one given that B had been picked already, and so on.

The expression for p3 will have six terms in the sum; one of these will be papb(pc+pd). The expression for p4 will have four terms in the sum (actually, it's the same term four times!)

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Edited by author 17.09.2022 01:09

When probability from 0~9 different, how to calculate the expected number???

Thank very much!!

I calculated probability of appearance of digit 0~9 as last digit and for first sample:
0: 0.925...
1: 0.009...
2~9: 0.008...
and expectation: 344.0203...

However, if i set probability from 0~9: all 0.1
the expectation becomes correct: 29.2896825397

I probably am missing something or misinterpreted something. Please, help me someone.

Many many thanks in advance.

[AC]
[Code Deleted]

Edited by author 01.11.2011 10:19

Edited by author 01.11.2011 13:10

Edited by author 01.11.2011 13:25