When the valve opens, the air flows towards less pressure. If we assume that pressure in the bigger balloon is higher than that of the smaller balloon, the air flows towards the smaller balloon.
However, in this case, it is assumed that pressure in the smaller balloon is higher, so air flows from the small balloon to the big balloon. Airflow continues as long as the pressure is higher in the small balloon. In the end, the balloons have the same pressure, and this means that neither of the balls are totally empty (unless they were empty at the beginning).
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