My answer to your question is probably much more detailed and philosophical than you may have expected! So, I’ll give you the short answer upfront and you can delve into the texts and theology that follow if and when you so desire! Bottom line: Praying for a good result on a CT or for the genetic health of a fetus would not be permissible according to the Mishnah because those prayers would not be able to have an effect on the outcomes, which have already been determined. However, if those prayers would help you deal with your personal health issues and anxieties, or if they would enable you to connect spiritually to God, to Jewish traditions, or to your community during your time of need, then, by all means, pray!
Here is my lengthy elaboration to this short response:
The Mishnah teaches that one is not permitted to pray for an outcome that has already been determined. This type of prayer is referred to as “tefillat shav” (“a vain prayer.”) For example, according to the Mishnah, if your wife is pregnant and you pray for a son, that is a “tefillat shav;” the gender outcome has already been determined and prayer cannot change the outcome. Another example of a “tefillat shav” is if you hear cries or screams coming from the direction of your town and you pray that the cries or screams are not coming from your house; again, the outcome has already been determined and prayer cannot change the outcome. Applying this Mishnah to your question, prayers for the genetic health of a fetus or for good results on a CT scan would be considered“tefillot shav” and would not be permissible.
However, if you look at the Gemara and other commentaries on this Mishnah,you will find that our matriarchs, Rachel and Leah, did in fact express gender preferences in their prayers. According to the Gemara, when Rachel was pregnant for the first time and Leah was pregnant for the seventh time, Leah prayed for her child to be a girl and for Rachel’s child to be a boy. Although the two sisters were already pregnant, Leah’s prayer turned out not to be ‘in vain’ after all; by a miracle, Leah’s male fetus was transformed into a girl (i.e. Dena) and Rachel’s female fetus was transformed into a boy (i.e. Joseph). The Gemara dismisses this apparent counterexample to the Mishnah by saying that we cannot refute the Mishnah’s law by means of an extraordinary miraculous event. Since miracles don’t happen in most cases, we can’t expect most prayers of this type to be effective.
Without meaning to sound disrespectful or heretical, I think that this section of the Gemara misses the point of prayer to some extent. Both the Mishnah and the Gemara seem to imply that for pray to be permissible, it has to be effective. But what does it mean for prayer to be effective? If, for example, I pray for a “refuah shelemah” (“complete healing”) for a loved one who is ill and she doesn’t survive her illness, does that mean that my prayer didn’t ‘work’? I think that we need to define the effectiveness of prayer in other ways, not just in terms of whether or not it ‘worked’ in causing a certain outcome to take place.
I do not believe that God is like a giant vending machine, into which we just have to put in the correct amount of money and then get the right product in return. I do not believe that if we say the right prayers and avoid the wrong prayers (such as tefillot shav) that God will respond in appropriate ways. I do not believe that we can control God. And, again, I do not mean to sound heretical, but I do not believe that God controls everything that happens in the universe. For one thing, God does not cause suffering. This may sound unconventional, but this is a legitimate Jewish belief. You need to ask yourself if you believe in a God who has the power to influence the outcome of chemotherapy or surgery, and if God will only do that if the right person recites the right words of prayer in the right language. You need to ask yourself if you believe in a God who would let a person die because a stranger, praying on his behalf, didn’t pray hard enough. I don’t believe in such a God.
So, if I believe that God does not directly control who gets sick and who gets better, then what are we doing when we pray to God for a “refuah shelemah”? What are we doing when we pray to God for a favorable outcome to a medical test or health crisis?
1) First of all, the term “refuah shelemah” “complete healing” does not necessarily mean that the person will literally gain back all of his physical health or stay alive. In fact, this type of thinking may not be realistic or even helpful or desirable for someone who is in pain. When we recite the “Mi She-berach”prayer, we use the terms, “refuat ha-nefesh,” “healing of the spirit,” and “refuat ha-guf,” “healing of the body.” Sometimes, it is realistic to hope for both types of healing (i.e. both the spiritual and the physical.) But, other times, it may only be realistic to hope for one of these types of recovery: either “refuat ha-nefesh”or“refuat ha-guf”. In some cases, it may only be realistic to hope and pray that our loved ones will find the peace of mind and the acceptance of illness that is part of spiritual healing, when physical health is not possible.
2) Secondly, when we pray to God for a “refuah shelemah,” we should realize that prayer is something that we do for ourselves. The Hebrew word for “prayer” is“tefillah.”“Tefillah” comes from the Hebrew verb “l’hitpalel,” which literally means “to judge yourself.” When we pray, we connect to the spiritual dimension of power to change ourselves. In other words, prayer is not a way to change God or to change the world; prayer is a way to change us. And when we change for the better, it is as if God has answered “yes” to our prayers.
3) Thirdly, when we pray to God for a “refuah shelemah,” we should realize that one of the main purposes of prayer is to put us in touch with God, but not in the way many people think. We should not approach God like a beggar asking for favors or a customer presenting God with a shopping list and asking God what it will cost. Prayer is not primarily a matter of asking God to change things. We cannot expect God to make us and to make those we love immune to disease. God cannot do that. We cannot ask God to weave a magic spell around us so that bad things will only happen to other people, and not to us. (That would be morally objectionable, in any event!) People who pray for miracles do not usually get miracles, any more than children who pray for I-pods, good grades, or boyfriends get them as a result of praying. But, people who pray for courage, for the strengthto bear the unbearable, for the ability to remember and appreciate what they haveleft instead of what they have lost very often find that their prayers are answered “yes.” God doesn’t send us suffering; but God gives us the strength to cope with suffering. God can renew our spiritual resources when they run dry. God doesn’t want us to be sick and God didn’t want our loved ones to be sick. God didn’t bring these problems into our lives and God can’t make these problems go away. But Godcan help us be brave when we are sick and frightened. And God can reassure us that we don’t have to face our pains and our fears on our own. We are never alone with God on our side.
4) Finally, beyond putting us in touch with God, prayer puts us in touch with Jewish traditions and with other people in the Jewish community. When we recite psalms or other Hebrew prayers of petition or request, using some of the same words that other Jews have used throughout Jewish history and throughout the world today, we are reminded that other Jews have had and continue to have the same concerns, values, dreams, and pains that we have today. When we recite those prayers in the context of a minyan or other congregational prayer context, we are reminded of this Jewish communal connection.
Even people who do not consider themselves to be ‘religious’ or ‘ritually inclined’ respond to a traditional religious wedding in the presence of friends and family, with familiar ceremonies performed and familiar words spoken, even though their marriage would be just as legal if it were performed in a judge’s chambers. We need to share our simchas, our joys, with other people. So too do we need to share our fears and our grief with other people. The Jewish custom of sitting shiva, the 7-day memorial period after a death, grows out of this need. When we feel so terribly alone, singled out by the hand of fate, when we are tempted to crawl off in a dark corner, we need to be reminded that we are part of a community, that there are people who care about us. Jewish traditions like shiva structure what we do, forcing us to be with people and to let them into our lives. When a mourner attends services to recite the Mourners’ Kaddish for 11 months after the death of a parent, she feels the context of a supportive, sympathetic community around her. She sees and hears other mourners and she feels less singled out by adverse fate. She is comforted by their presence, by being accepted and consoled by the community during her bereavement.
Prayer, whether it is the Mourners’ Kaddish or the “Mi She-berach,” can redeem each of us from isolation. It assures us that we need never feel alone or abandoned. Prayer lets us know that we are part of a greater community and a greater reality, with more depth, more courage, and more of a future than any of us, as individuals, could have by ourselves.
The widow who thought, after the death of her husband, “What do I have to live for now?” yet, in the course of ensuing weeks, found reasons to wake up in the morning; the man who lost his job and thought that he’d never be able to start over again, but started over again any way, where did they get the hope and strength to begin again? In part, the widow and the unemployed man became stronger and more hopeful because they prayed in the context of a concerned community, people who made it clear to them that they cared, andfrom the knowledge that Judaism would always be there for them and God would always be right by their side. But also, the widow and the unemployed man may have realized that they had more internal strength and courage than they ever knew themselves to have had. Their prayers may have helped them tap into hidden internal reserves of faith that were unavailable to them before their experiences of loss.
Whenwe open our hearts in prayer for ourselves or for someone we love who is ill or whose future health is uncertain, we may not be able to get a miracle or avert a tragedy. But, we may discover the people around us, and we may discover the comfort and familiarity of Jewish traditions available to us, and we may discoverGod beside us, and we may discover the strength within ourselves to help us deal with our anxieties and fears.
In these respects, our prayers are always effective.
By all means, pray for good results on a CT scan or for the genetic health of a fetus if you think that the prayers will help you deal with your internal anxieties and fears, connect with God, with Jewish traditions, with the Jewish community, or with other people who stand beside you in prayer in body or in spirit.
 Mishnah B’rachot 9:3 and the Babylonian Talmud, B’rachot 54a
 “ha-tzo-ek l’she-avar” (“one who screams out about something that has already happened in the past”)
 The Hebrew word, “levatala,” that you used in your question may also be translated as “in vain.” However, the halachic phrase for a “vain prayer,” that pertains to the context of your question is“tefillat shav.” “Shav” is the same Hebrew word for “vain” that is used in the 3rd commandment, “Do not take God’s name in vain.” The Hebrew word, “levatala” is used in a different halachic context, that of “b’racha levatala,” “a blessing that is said in vain.” This phrase refers to the recitation of a blessing pertaining to the eating of a food or the performance of a mitzvah in a situation when you do not actually eat the food or perform the mitzvah. For example, some say that you should not recite the Hebrew words, “Baruch Atah Adonai Eloheynu Melech ha-olam ha-motzi lechem min ha-aretz”(“Blessed are You our God Ruler of the universe who brings forth bread from the ground.”) unless one is actually going to eat a piece of bread following the recitation of the blessing. It is for this reason that some people substitute the word “HaShem” for God’s Hebrew name, “ Adonai,” when teaching another person to recite a blessing or when reading about the recitation of a blessing in a book (or in the response to a Jewish Values Online question!), so as not to recite a blessing in vain, “bracha levatala.”
 Midrash Sechel Tov, Targum Yonatan on Parashat VaYetze, etc., etc.
 And, if you’re not satisfied with that means of dismissing the Leah/Rachel counterexample, the Gemara provides another reason for dismissing it, by saying that the gender is not determined within the first 40 days of conception and that Leah’s prayer was uttered within the first 40 days of conception
The question asks when it is permissible to ask for God’s intervention, but suggests that a limit might exist after certain points in time, such as between the time when a CT scan is taken and when the patient receives the results. The answer depends on one's motivation, whether you are asking for God to upend the order of nature or asking for God’s compassion in a difficult situation.
Prayer is the “service of the heart” (Maimonides, Hilchot Tefila 1:1), the way we serve God with all dimensions of our heart. The question reflects a situation described in the Mishnah (Berachot 9:3), “If one cries out to God over what is past, his prayer is in vain,” implying that the intent of the prayer is to undo events that have already taken place. Examples offered by the Mishnah include praying that a child already in utero be of a certain gender, or praying that an alarm that is already sounding is not sounding in one’s home. The child already has a designated gender, the alarm is for an event that has already taken place – prayer will not change what is already a fact. Such a prayer is known as tefilat shava, a prayer in vain.
Is it true, however, that we do not pray for God’s help in situations which seem as if they are already certain? We assert in the daily Amidah, individual prayer, that God hears prayers, acts with compassion, and does not turn us away empty-handed. Traditional prayerbooks note that one may insert a private prayer on behalf of oneself or others at this point in the service, just as you can in the earlier prayer for healing. Similarly, when we stand before the ark we often recite the MiSheberach prayer for healing and mention the names of those for whom we wish healing of body and soul. Many of those individuals have already received a diagnosis, nonetheless we pray that God might act with compassion toward them and give them a perfect healing of body and soul.
Our prayers are not intended to reverse nature or to replace medicine. Rather they express the yearnings of our heart. Prayer gives voice to our fears, concerns, and loneliness. Prayer directs the attention of our heart, our community and of God to the place where compassion and attention are needed – the person who is ill, alone, in need. The Talmud teaches that bikkur holim, visiting the sick, removes 1/60th of the illness. That does not suggest that if 60 people visited one who was ill they would no longer be ill, rather it acknowledges that prayer, attention, and care lift the spirits of one who is ill and supports them in their own journey.
There is a distinction between a beracha levatala, a blessing recited by mistake or one that is recited but not acted upon; a tefilat shava, a prayer that asks God to reverse or change the order of nature; and a bakashat rachamim, a request for God’s compassion. The Talmud (Baba Metzia 59a) recounts the teaching of R. Elazar:
Since the destruction of the Temple, the gates of prayer are locked, for it is written, Also when I cry out, he shuts out my prayer (Lamentations 3:8) Yet though the gates of prayer are locked, the gates of tears are not, for it is written, Hear my prayer, 0 Lord, and give ear unto my cry, hold not thy peace at my tears.(Psalm 39:13)
Our prayers for Divine compassion, even in the face of test results, can express our soul’s cry for help in the face adversity and uncertainty. Such prayers are always appropriate.
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