try to do simulation with sapply

I am trying to simulate how soon the product of uniform(0,1) random number will be < exp(-3). So I write like below

n = c(10^2, 10^4)
p <- 0:6

df2 <- data.frame(expand.grid(n, p))
names(df2) <- c("n", "p")

find.p <- function(n, p){
  v <- c()
  for(i in 1:n){
    product <- 1
    j <- 0
    while(product > exp(-3)){
      product <- product * runif(1)
      j <- j + 1
    }
    v[i] <- j - 1
  }
  v.re <- table(v)/n
  print(p)
  return(v.re[names(v.re)==p])
}

df2$pr <- sapply(df2$n, find.p, p=df2$p)

I record when the product < exp(-3). I want to know what is the probability that the product < exp(-3) at 0:6 and redo 100 and 10000 times, expecting convergence. I try to do it with sapply, but I always got wrong result the distribution should be approximate to Poisson (3).

Any suggestion would help me a lot!