Why did the South believe it could win the war despite numerical disadvantages?
The South knew it was at a numerical disadvantage, and a severe one, in terms of manpower to fight a war, as the North had more than twice the population. The South, however, knew it had a leadership advantage. It had generals like Robert E. Lee. Many of the United States' highest ranking officers were Southerners, and seven of the eight military academies in the country were located in the South. Initially, relying on better military leadership did yield advantages: the North was hampered by less competent generals, the South won battles, and it took some time before Lincoln was able to get in place the kind of generals he needed.
A far more critical factor, however, was the failure to persuade England to intervene in the war on the side of the South. Confederate President Jefferson Davis felt certain that by withholding the cotton the English needed for its mills, England would be forced into the war. The South knew from the start it needed this support; as we know, however, England remained neutral and the South lost the war, remaining part of the United States.
The South believed that it could win the war because it had its own advantages. Perhaps the two most important were its fighting spirit and its foreign relations.
The South felt that its men were better suited to fighting than Northerners. A disproportionate number of Army officers were from the South. Southerners rode horses and hunted much more than Northerners. This made the South feel its men would simply fight better than the Northerners.
The South felt that its foreign relations would help it win the war. It felt that cotton would accomplish this. The South believed that countries like France and especially England would need Southern cotton so much that they would side with the South and pressure the North to let the South be independent.
For these reasons, the South felt that it could overcome its disadvantages and win the war.