Discrepancy Between 2 Different Ways Of Calculating - What's Wrong, Please?

I am trying to answer the following question: if an event is expected
to happen n times (e.g. twice) in a 3 year period, what is the
probability of it happening 4 or more times?

Here's a Poisson distribution function I wrote:

(%i46) poisso(x,lam) := (i: x, sum((%e^(lam * -1) * lam^i)/i!, i,0,i));


lam (- 1) i
%e lam
(%o46) poisso(x, lam) := (i : x, sum(----------------, i, 0, i))
i!


(%i40) poisso(4,2),numer;
(%o40) 0.94734698265629

So, there's about a 5% chance of it happening 4 times. This can be
checked with the Poisson Distribution calculator at
http://www.ciphersbyritter.com/JAVASCRP/BINOMPOI.HTM .

I also wanted to be able to calculate it day by day. Assuming that 3
years = 1095 days, if the event is expected to happen twice, then its
daily probability will be 2/1095, or 0.001826. Using the binomial
calculator at http://www.ciphersbyritter.com/JAVASCRP/BINOMPOI.HTM ,
with p=0.001826, n=1095 and k=4, we again get an answer of about 5%.

I then wrote a monte-carlo simulator (below) - and it consistently
gives answer of around 14-15%. I cannot see where the discrepancy is
arising from - can anyone spot it for me, please?

In this Maxima program, 1000 simulations of 3 year periods (1095 days)
are done. A day is "counted" if a random integer in the range 0..1095
equals 0 or 1 (which leads to an expected value of 2 counts in the 1095
days), and an "event" is counted if 4 or more counts occur in the 1095
days.

(events: 0,
for outer: 1 thru 1000 do
(count: 0,
for inner: 1 thru 1095 do
if random(1095) <= 1 then
count: count + 1,
if count >= 4 then
events: events + 1),
print("Event happened 4 or more times in", float(events/10), "% of the
tests"))$

(I also wrote it in visual basic, and got the same answer):

Dim count, events As Integer
Dim level As Double
level = 2/1095
events = 0
For outer = 1 To 1000
count = 0
For inner = 1 To 1095
If Rnd() < level Then
count = count + 1
End If
Next
If count >= 4 Then
events = events + 1
End If
Next
Messagebox Cstr(events/10) & "%, (" & Cstr(events) & ")"

.