In The Hunger Games, when does Katniss realize that she has a chance of winning?

This is an interesting question to consider. If we look at
the text carefully, we can see that the first time Katniss mentions the prospect of
winning comes after Rue is dying from the spear wound in Chapter 18. Katniss has of
course by this stage blown up the food at the starting point, which is a major
accomplishment. In response to Rue's command that she "has to win" the Hunger Games,
note how Katniss responds:

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I'm going to. Going to win for both of us now," I
promise.

This is perhaps the
first recognition that Katniss shows of the possibility of her victory, and also this
comment shows great determination. At this point, of course, the contenders have been
thinned down significantly, so Katniss does have a real chance. After this stage in the
book, as others die and Katniss remains, obviously her chances of winning become greater
and greater.