PoissonBinomialDistribution[ plist : { __?NumericQ } ] := With[
{
  n = Length @ plist,
  c = Exp[(2 I [Pi])/(Length@plist + 1)]
}
,
ProbabilityDistribution[
    Re[ 1/(n + 1) Sum[c^(-l k) Product[1 + (c^l - 1) plist[[m]] , {m, 1, n }], {l, 0, n}] ]
    ,
    {k, 0, n, 1}
]
] /; AllTrue[ plist, 0 <= # <= 1& ]

dist = PoissonBinomialDistribution[ {0.04, 0.07, 0.07} ];

Probability[ k == 3, k [Distributed] dist ]// PercentForm

SuccessProbability[plist_,k_] := ...

SuccessProbability[{0.04, 0.07, 0.07},3]

0.0196

SuccessProbability[plist_, k_] := Probability[k == 3, k [Distributed] (With[{n = Length@plist, c = Exp[(2 I [Pi])/(Length@plist + 1)]}, ProbabilityDistribution[Re[1/(n + 1) Sum[c^(-l k) Product[1 + (c^l - 1) plist[[m]], {m, 1, n}], {l, 0, n}]], {k, 0, n, 1}]] /; AllTrue[plist, 0 <= # <= 1 &])] // PercentForm