I agree with you that statement (1) is sufficient...since the number of red balls relative to the white and black is greater than the ratio of white balls to black and red, it follows that a red ball will have a greater probability of being chosen.
However, statement (2) only gives us b-w>r (or b>r+w). This tells us that black will have the highest chance of being chosen (higher than red and white combined), but between red and white, we cannot tell which is higher. For example, if there are 10 balls total, we could have 6 black, 3 red, and 1 white in which case red would have a higher probability. But we could also have 6 black, 3 white, and 1 red. Statement (2) would still be satisfied, but now white would be a more likely outcome.
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Source: Beat The GMAT — Data Sufficiency |
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givemeanid
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The question is asking is r > w?
1. r/b+w > w/b+r
br + rr > bw + ww
b(r-w) > ww - rr
b(r-w) > (w+r)(w-r)
r-w > k(w-r) -> since b and (w+r) are both positive, k = (w+r)/b = positive
Now, if r < w, left side is -ve and right side is +ve. The inequality doesn't hold.
So, r > w.
SUFFICIENT.
2. b - w > r
INSUFFICIENT.
1. r/b+w > w/b+r
br + rr > bw + ww
b(r-w) > ww - rr
b(r-w) > (w+r)(w-r)
r-w > k(w-r) -> since b and (w+r) are both positive, k = (w+r)/b = positive
Now, if r < w, left side is -ve and right side is +ve. The inequality doesn't hold.
So, r > w.
SUFFICIENT.
2. b - w > r
INSUFFICIENT.
So It Goes
