How do Macbeth's erroneous interpretations of the three apparitions (and the predictions) in Act Four, scene one, affect the outcome of Shakespeare's Macbeth?
In the witches' second set of predictions in Shakespeare's Macbeth, Macbeth sees what he wants to see. He is easily tricked by the Weird Sisters into believing that he is invincible. This "false sense of security" will be his downfall.
In Act Four, scene one, the new predictions are generated to get Macbeth not only more deeply dependent upon what the witches' tell him, but at Hecate's orders, to win his soul completely to the power of evil: Hecate also wants some of the credit for Macbeth's deterioration and destruction.
When the witches tell Macbeth to "Beware Macduff," he simply thinks they are telling him something he already knows. This will affect the story in two ways. In his paranoia, he will decide that to be safe, he should kill Macduff. (However he hesitates and by the time he gets around to it, Macduff has left the country to see Malcolm.) Macbeth has Macduff's entire family and household murdered. Of course, this act enrages Macduff to the point that no one else had better get to Macbeth before he does when they invade Scotland—with a score to settle, Macduff will avenge his family, and probably, too, the much-loved Duncan.
When the witches' second prediction is delivered, it tells him that only one kind of man can kill him:
...for none of woman born
Shall harm Macbeth. (89-90)
The apparition shows a bloody child. Macbeth believes that this prediction means no man can harm him—because all men are born of a woman. However, even believing he needs not fear Macduff, he decides he will take steps just to make sure.
This, of course, foreshadows Macbeth's death at the hands of Macduff. The bloody child in the apparition is Macduff as an infant: he was born by Cesarean section, not by natural childbirth. Not knowing this, Macbeth believes he cannot be defeated. It will be at Macduff's hands that he will die.
The final apparition is that of a crowned child holding a tree branch in his hand. The prediction is:
Macbeth shall never vanquish’d be until
Great Birnam Wood to high Dunsinane Hill
Shall come against him. (103-105)
Macbeth does not realize that the crowned child represents Malcolm. The tree branch in his hand symbolizes the branches that Malcolm will order his men to cut down and use to hide their true numbers.
Let every soldier hew him down a bough,
And bear't before him: thereby shall we shadow
The numbers of our host, and make discovery
Err in report of us. (V.iv.6-9)
Because Macbeth is sure the woods and the hill cannot move, he sees this last prediction as just one more guarantee that he cannot be beaten. However, in that Macbeth does not believe he can be defeated, the crowned child means nothing, and the tree branch has no significance for him. However, Malcolm will be successful in masking his numbers from Macbeth's spies; Macbeth will have no idea how large the force is that attacks his castle. And it is not until the trees look like they are moving (with the cut branches) that he begins to suspect the witches of telling him half-truths.
Because Macbeth underestimates the significance of the apparitions, and because he takes the predictions too literally, he is ultimately defeated and killed. His foolish belief that he "bears a charmed life" lulls Macbeth into thinking that he has nothing to fear. As a soldier, this kind of casual behavior can be deadly: and it is for Macbeth in the end.