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Our walk is a walk of faith, not of "luck". The better response would be to offer up prayer for the person.
I would say that the key verse with regard to this question is Romans 8:28. As I see it, this verse does not deny the influence of the fallen world, our sinful flesh, and the devil in causing events that may seem adverse to happen to us. (To me, Jesus also seems to be saying this in Luke 13:1-5.) However, God is in control, and even such "misfortunes" happen only with His permissive will (Matthew 10:29). And God is capable of making those events (as unpleasant or even disastrous as they may seem to us at the time) to work together for our ultimate good, as long as we remain faithful to Him. From that standpoint, it would be appropriate (in my opinion) to ask God's care and blessing on someone, but not to wish them good luck, since that (to me) would be implying that things happen that are outside of God's control.
The notion or concept of "luck" (whether good or bad) seems to have originated with the Greeks and Romans. The Greeks had a popular goddess named Tyche (Roman equivalent to Fortuna) who was associated with luck, providence and fate. Today these goddesses could be no more than demonic spirits, if they still exist. Rather than believing in the favor or disfavor of past pagan entities (whether intentionally or not) we should as Christians instead believe in the Lord's Prayer where it says "and lead us not into temptation and deliver us from evil." If we are led by Christ we are not subject to the whims of luck since "we know that all things work together for good to them that love God, to them who are the called according to his purpose" (Rom 8:28).
I really don't like answering a question with a question but here goes; Who said we shouldn't say "Good luck" to people? Perhaps if they are not a Christian and by one saying "Bless you" or May God bless you" or whatever seem religiously appropriate may actually turn them off. Our talking to people in a language they understand may actually bring them to God in ways religious verbiage may never do. Jesus spoke to the people of His time in a manner of which they understood, Paul and James did the same so why should we be any different. I do not believe in luck but will say to people be they Christian or not "Good luck" because for me I'm not concerned with what people think and they do not know what is in my heart when I trust God for their good. In many cases it is only a matter of words and how one chooses to define them IE; I wish can = I hope. Don't we for the most part tell our children to make a wish and blow out the candle on their birthday cake. That is not using faith, it may raise another question if it is not already on here: Should Christians celebrate birthdays? We can find some places in scripture where luck was referred to or something close to luck; Ecclesiastes 9:11 Again I saw that under the sun the race is not to the swift, nor the battle to the strong, nor bread to the wise, nor riches to the intelligent, nor favor to those with knowledge, but time and chance happen to them all. (Time and luck happened to them all?) Luke 10:31 Now by chance (luck) a priest was going down that road, and when he saw him he passed by on the other side. 2 Samuel 1:6, Acts 27:12 Genesis 30:11 And Leah said, "Good fortune (Good luck) has come!" so she called his name Gad. There are other references that can be found throughout scripture and the one thing that can be found in scripture that is consistent with God and that is relationship. I believe God is more concerned with our relationship with him and others that He is not focused on our verbiage. If I can gain and maintain a relationship with someone be they Christian or not and do my best to show them the love of God I have done more than just use appropriate "Christian verbiage". Paraphrased; Jesus did say that if you find someone hungry and say be blessed and send them on their way as you have done to them you have done to Him James indicates that you say you have faith but I will show you my faith by my works, so perhaps God is more concerned with what we do more than what we say or how we say it. We can tell people how much we love them but they don't believe us until they see how much we love them. I would consider being more concerned with how much we love people and show them the love of God over what is "Correct or appropriate Christian" verbiage in order to make a difference in their life. Many times I have been told "Bless you brother" and then get a knife in the back but would prefer "Hey good luck man, I'm praying for you" Some phrases one may consider using are; Good luck, I'm praying for you I trust all will go well I wish you the best I'm sure it will be fine There are other ways of expressing our desire for good results or good things in the lives of others that communicates sincerity which in my opinion is what people are looking for. 1 Cor 12:31 But earnestly desire the higher gifts. And I will show you a still more excellent way. Desire the HIGHER gifts 1 Cor 13:13 So now faith, hope, and love abide, these three; but the greatest of these is love.
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