God determines everything that will come to pass. If you pray that your dog will live, and your dog dies, God has answered your prayer in the negative.
Sometimes it is not that cut and dried.
1. Ambiguities as to what Ought to be Done
To take one example, a young person may pray for a long time to find a suitable spouse, but no spouse may appear. This may be God answering in the negative, or simply God encouraging the young person to pray harder! It's really impossible to determine the answer to that question simply by looking at the results.
Lesson: we should not try to look to Providence alone to determine how we should proceed for the future. If a young person came to me asking for advice about whether to continue the quest for marriage or to give up and remain single, I'd need to know a lot more than, "I've been praying about this for a long time now."
2. Ambiguities as to Bad Things Happening
a. Job's Friend's Syndrome
Sometimes God sends bad things into our lives to chastise us for wrongs that we have done, so that we may learn to do what is right. It's unpleasant at the time, but good for us in the long run (just like corporal discipline of children). When bad things happen, we should seriously consider whether God is disciplining us, and examine our lives for ways to improve.
On the other hand, that is not always the case. So, we should be careful not to be Job's judgmental friends to others. This is not an excuse to avoid self-examination, this is an admonition to avoid a judgmental spirit. We cannot and should not assume that the twin towers disaster killed the 3,000 least holy people in Manhattan a half decade ago. We cannot and should not assume that their widows and orphans were the most wicked wives and children of the greater NYC region. All have sinned, and unless we repent we will also perish.
b. Satan did it Syndrome
There is a reverse problem to the Job's friend's syndrome, which I call the "Satan did it" syndrome. When bad things happen, sometimes it can be persecution from wicked men or even wicked angels. It's a bit presumptuous to think that the Prince of darkness has personally found time to mess with one's life (after all, he is not omnipresent, omnipotent, etc.), but nevertheless sometimes the forces of evil stand opposed to us and cause bad things to happen to us.
On the other hand, when something we thought was a good idea doesn't happen, simply asserting that Satan interfered can lead to a dangerously egotistical mindset. "We tried to plant a church in Boise, ID, but Satan snuffed it out," the person might say. Maybe it wasn't Satan! Maybe God simply decided that Boise has enough of a witness, and God was closing the door there. Blaming Satan, you see, is often simply a way of reinforcing one's notion that one knows best.
Such a person will have trouble learning from God's corrections. If they lose their health, they simply say that Satan is messing with them. If they lose their money, it's Satan trying to stop them. If their car breaks down on the way to some activity, it is proof how good that activity was to which they were headed!
And it's doubly foolish. First of all, it can result in blasphemously attributing the chastisements of God to Satanic interference. But on top of that, God can use Satan to chastise God's people. Job's losses were not primarily discipline for Job, but Job was not sinless. Job was a god-fearing man, but he was not perfect. Read to the end of the book of Job and you'll see.
But Job is not even the best example here. Recall:
1 Timothy 1:20 Of whom is Hymenaeus and Alexander; whom I have delivered unto Satan, that they may learn not to blaspheme.
You see, God can strike a straight blow with a crooked stick. Even though Satan did not afflict Jim and Alex out of love of God, God used Satan's affliction for their good. If they simply said to themselves, I must be doing something good, since Satan's persecuting me, they'd be stuck in the "Satan did it" syndrome, and unable to learn not to blaspheme.
3. Ambiguities as to when Good Things Happen
The good things are easier to handle. It's great when the silos are full of corn, the wine cellars are full of wine, the bank account is full of cash, and the table is full of children. Thank God when you receive such blessings.
But still be careful to examine yourself. Recall that you are just a visitor here. This life is not all there is. You may receive the good things of this life from God, but there is a life to come. Do not be like the rich man, who put off his concern for the afterlife until he was already in the pit. Be like Job: hold what good things God has given you loosely enough that you can be ready to say, "the LORD gave, and the LORD hath taken away; blessed be the name of the LORD." That's not easy.
More than that, don't assume that because everything is going well, God is happy with you. Recall those who robbed David:
1 Samuel 30:16 And when he had brought him down, behold, they were spread abroad upon all the earth, eating and drinking, and dancing, because of all the great spoil that they had taken out of the land of the Philistines, and out of the land of Judah.
They were feasting and rejoicing. But destruction was looming.
1 Samuel 30:17-19
17And David smote them from the twilight even unto the evening of the next day: and there escaped not a man of them, save four hundred young men, which rode upon camels, and fled. 18And David recovered all that the Amalekites had carried away: and David rescued his two wives. 19And there was nothing lacking to them, neither small nor great, neither sons nor daughters, neither spoil, nor any thing that they had taken to them: David recovered all.
What is the bottom line? We should be cautious about reading Providence. God deals in ways that are often obscure. Joseph may have seen only his brother's jealousy during his trip to Egypt, or he may have seen correction to his own pride. But even that was not the big picture. The big picture was that God was working out the good of his chosen people, showing his love for them.
Whatever adversities may find you, whatever circumstances may happen to you, if you do not trust in Christ alone for salvation you should tremble with fear, but if you believe on the Lord Jesus Christ for salvation take comfort in God's assurance that:
Romans 8:28 And we know that all things work together for good to them that love God, to them who are the called according to his purpose.
Praise be to our Glorious King!
UPDATE: S. Todd Young has something similar to say, especially about Joseph (link).