Genesis 17:17 then abraham fell upon his face, and laughed, and said in his heart, shall a child be born unto him that is a hundred years old? and shall sarah, that is ninety years old, bear?
The Old Testament character whose name means laughter is Isaac. His name was given to him because his mother Sarah laughed when she heard she would have a son in her old age.
Abraham loved Sarah because she was his wife. Abraham loved Hagar because of Ishmael. So Abraham loved both. That is why it was very difficult to choose between them if there had been no Divine intervention.
God told Abraham that he would have a son at his old age.
Abraham was exceedingly happy when Ishmael was born and wished that Ishmael would be the child of promise, showing little faith in God's abilities. But God corrected Abraham and told him that his son Isaac will fill that role. He had other plans for Ishmael:
Hagar and Ishmael were sent away because of tensions between Hagar, the mother of Ishmael, and Sarah, Abraham's wife, after the birth of Isaac. Sarah felt threatened by Ishmael's presence and demanded that Abraham send them away to ensure that Isaac would inherit his lineage without competition. God reassured Abraham about the situation, instructing him to listen to Sarah, promising that Ishmael would also become a great nation.
Abraham laughed in happy wonderment that he and Sarah, both aged, would soon have a son (Genesis 17).See also the Related Link.More about Abraham
Abraham did not believe what God told him. God told Abraham that he and his wife Sarah would have a baby. But Abraham thought it was impossible because Sarah was in her 90s. So Sarah told Abraham to sleep with with Hagar there servant, there child was Ishmael. Later, Sarah changed her mind and made Abraham send Hagar and Ishmael away. Some time after this Abraham slept with Sarah and she became pregnant with Isaac.
God promised Abraham and Sarah that they would have many descendants and that they would be the ancestors of a great nation. He also promised them the land of Canaan as an everlasting possession for their descendants.
Yes, when Sarah saw Ishmael teasing Isaac, she asked Abraham to send Ishmael and his mother, Hagar, away. At first, Abraham did not want to, but God spoke to him, telling him to listen to Sarah's voice. This was the beginning of a fulfillment of a prophecy. Isaac would be the one through whom God's purpose would be fulfilled. The one through whom Messiah would come (Genesis 21:8-13)(Genesis 17:15-21/Genesis 25:5-9) who would provide a hope for ALL mankind, and benefit any who chose to accept it (Genesis 12:3/Galatians 3:14).
Sarah tried to make God's promise to Abraham come to pass, by having him take her maidservant as a second wife, as per the custom in those days. The promise was that Abraham would have descendants.See also:More about Abraham
I believe the short answer is unbelief. God has promised a son to Abraham, and that this son would come through Sarah, even though Sarah was advanced in years. Either Sarah didn't believe God at all, or she didn't believe the promised child had to come through her. Isaac was the son born of the promise, through Sarah as God had promised. The first comment is quite correct. It has also been discovered that this was an ancient practice and so her actions were not without precedent even though prompted by unbelief. Discovery of this ancient custom underlines the social 'fitness' of the Genesis record of history as opposed to the Documentary view which puts the composition much later when such social customs would have been unknown.
Ishmael was driven out because Sarah, Abraham's wife, was jealous of him being the son of Hagar, her maid. Sarah asked Abraham to send Hagar and Ishmael away to ensure that Isaac, her own son, would be the sole heir.