Geodesic
On a problem of choosing the order of dependent trials
Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 4, pp. 752-755

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A sequence of trials with numbers $1,\dots,n$ is considered. The order of trials is given by a permutation $\tau_n=(i_1,\dots,i_n)$. The probability $p_{i_k}(m)$ of the success in the $k$-th trial depends only on the total number $m$ of successes in $k-1$ previous trials. Conditions are given for $\tau^*=(n, n-1,\dots,1)$ to maximize the expected total number of successes. 
@article{TVP_1972_17_4_a14,
     author = {I. B. Gercbah},
     title = {On a problem of choosing the order of dependent trials},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {752--755},
     publisher = {mathdoc},
     volume = {17},
     number = {4},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1972_17_4_a14/}
}
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I. B. Gercbah. On a problem of choosing the order of dependent trials. Teoriâ veroâtnostej i ee primeneniâ, Tome 17 (1972) no. 4, pp. 752-755. http://geodesic.mathdoc.fr/item/TVP_1972_17_4_a14/