Compute conditional probability for Poisson binomial distribution Consider $X=Y_{1}+\cdots+Y_{n}$, where $Y_{1}, \cdots, Y_{n}$  are $\mathrm{n}$ independent Bernoulli random variables  with $Y_i\sim Bernoulli (1,p_i)$,  $i=1,2,\cdots,n$.  Then $X$ has a so-called Poisson-binomial distribution of parameters $p_{1}, \cdots, p_{n}$ and $n$.
Suppose $n=20$ and $p_i=0.5+i/50$,  $i=1,2,\cdots,20$. How to compute the conditional probability of $P(X=i |X\le 5)$, $i=1,2,3,4,5$, by formula derivation or software simulation (R or Python) ?
So far, I  try my best to have the following results.
$$
P \triangleq P(X=i |X\le 5) = \frac{P(X=i,X\le5)}{P(X\le 5)}
\overset{i=1,\cdots,5}{=} \frac{P(X=i)}{P(X\le 5)}
$$
Then I resort to the library(poisbinom) package in R to compute the conditional probability P
# 20 Bernouilli r.v.s
n = 20; ii = 1:n; pp = 0.5 + ii/50
library(poisbinom)
numerator = dpoisbinom(1:5, pp) #P(X=i), i=1,2,3,4,5
denominator = ppoisbinom(5, pp) #P(X<=5)
p5 = numerator/denominator
round(p5,5)
# [1] 0.00001 0.00040 0.00742 0.09509 0.89707

However, I am not sure whether my results are correct.  Further, is there any other ways [formula derivation or software simulation (R or Python)] to calculate the above conditional probability?

Is there a reason you did not mention the small possibility $X=0$?
That being said, your rounded numbers look correct.  Here is a recursion producing the same results (including $0$, which has a conditional probability about $0.000000207$)
maxi <- 20
limitn <- 5
bernprob <- function(n){ 1/2 + n/50 }
probmat <- matrix(0, nrow=maxi+1, ncol=maxi+1)    # offset to include 0
probmat[1,1] <- 1
for (i in 1:maxi){
  probmat[i+1,] <-  (c(probmat[i,], 0) * (1-bernprob(i)) + 
                     c(0, probmat[i,]) * bernprob(i)    )[-(maxi+2)]
  }   
condprob <- probmat[maxi+1, 1:(limitn+1)] / sum(probmat[maxi+1, 1:(limitn+1)])
names(condprob) <- 0:limitn
condprob  

#            0            1            2            3            4            5 
# 2.070919e-07 1.338460e-05 4.012590e-04 7.423162e-03 9.509096e-02 8.970710e-01

