Is it possible for a match to be tied after applying the Duckworth-Lewis method?
Are there notable instances of this occurring?
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Yes, it's possible for a match to get tied after applying the D/L method.
Some matches have ended in a tie this way, an example being the South Africa vs West Indies group stage match in Champions Trophy 2013.
Is it possible for a match to get tie even after applying D/L method?
Why not? Suppose a team scored 280 runs in 50 overs. Team batting second gets target of 271 in 45 overs. If they score 270 after 45 overs, the match will be tied. Another simple example: A match is affected by rain before the match is started. So when the match is started it is reduced to 40 overs per side. If both teams score same runs the match will be tied as per D/L method.
Is there any match tied after applying this tie breaker?
Firstly, D/L method is not a tie breaker. It is a method to calculate the target for the team batting second in the rain/weather affected matches.
There are some ODI matches which were tied after the D/L method was applied. The match between Ireland v Pakistan is another example of a tied match using D/L method. Here is the list of tied ODI matches. You can search other matches yourself where D/L method was applied.
Regarding T20I matches there is no match which was tied after D/L method was applied*. You can see the list of tied T20I matches here.
* As on 2014-02-10
Of course, yes.
The best example for your question is match between South Africa VS Sri Lanka in 2003 world cup which was actually tied game under D/L method which eventually made South Africa to get eliminated from the tournament in group stages.