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!)