No, this isn't the reason we pray for our leaders and for peace -- not at all. If this was the reason we prayed, we might as well quit because it would be useless.
We pray this way first of all because God has commanded us to pray for our world and its leaders. Every day, our leaders face complex decisions that are almost overwhelming, and they need God's guidance if they are to make them wisely. The Bible says, "I urge, then, first of all, that requests, prayers, intercession and thanksgiving be made for everyone -- for kings and all those in authority, that we may live peaceful and quiet lives" (1 Timothy 2:1-2).
But we pray, too, because God is at work behind the scenes of history -- even if we don't always see it. I agree with you up to a point; it's easy to wonder if our prayers do any good because so little seems to change. But have you ever asked yourself what the world would be like if we stopped praying? I'm convinced we'd be in even worse shape.
Today, our nation celebrates the annual National Day of Prayer. Take time to pray for our nation and its leaders -- and for the leaders of other nations also. God has promised, "If my people, who are called by my name, will humble themselves and pray and seek my face and turn from their wicked ways, then will I hear from heaven and will forgive their sin and will heal their land" (2 Chronicles 7:14).