While the philosophy of religion is a sub-field of philosophy, the issue you are referring probably fits better under psychology. In psychology there is a whole area devoted to religion, spirituality, and health, and even though your question does not necessarily involve health, it's specifically about intervention which is at the heart of much of the research. In particular, there are numerous studies which try to claim that there is scientifically verifiable evidence as to the existence of God's intervention, as instigated through means such as intercessory prayer (this is, incidentally, one of the areas I studied and wrote a thesis on in college).
The crux of the issue is the lack of a plausible causal mechanism if intervention is actually occurring. There is also the issue of a mixed agenda from a number of religious scientists who want to prove the existence of God (I've read both Christian and Islamic scientists who seem to have such an agenda), however from reading their papers it seems that they generally do not realize that even if a significant effect of intercessory prayer on health outcomes was found and could be easily manipulated on a consistent basis, at best this would prove that there is some as of yet unknown mechanism (not even necessarily sentient) that mediates personal wishes for others. In no way would any particular religion or deity be affirmed, much less the content of their respective religious texts. And even if it somehow did prove a deity existed, Andrade & Radhakrishnan (2009) remind us of a sobering fact: if research proves a link between intercessory prayer and health outcomes, doesn't that suggest that Gods ways can be manipulated or his behavior made statistically predictable? I'm not sure that's a claim the religious would want to uphold.
I think the above explanations should sufficiently answer and address the issues in the hypothetical you set up in your question. As for your over-arching (title) question, "Does God intervene in our lives?", ignoring the implicit assumption that God exists, science says "We have no evidence for it so we presume not until proven otherwise," but you can feel free to search through the academic literature for yourself, as some (very few) people seem to be convinced that we do have enough evidence. With regard to intercessory prayer in particular, I can give several paper references as a starting point. You may, however, be interested in looking at other types of intervention as well.
Andrade, C., & Radhakrishnan, R. (2009). Prayer and healing: A medical and scientific perspective on randomized controlled trials. Indian Journal of Psychiatry, 51(4), 247-253.
Benson, H., Dusek, J., Sherwood, J., Lam, P., Bethea, C., Carpenter, W., Levitsky, S., Hill, P., Clem, D., Jain, M., Drumel, D., Kopecky, S., Mueller, P., Marek, D., Rollins, S., & Hibberd, P. (2006). Study of the therapeutic effects of intercessory prayer (STEP) in cardiac bypass patients: A multicenter randomized trial of uncertainty and certainty of receiving intercessory prayer. American Heart Journal, 151(4), 934-942.
Breslin, M. J., & Lewis, C. A. (2008). Theoretical models of the nature of prayer and health: A review. Mental Health, Religion & Culture, 11(1), 9-21.
Byrd, R. (1988). Positive therapeutic effects of intercessory prayer in a coronary care unit population. Southern Medical Journal, 81(7), 826-829.
Dorn, J. M. (2006). Intercessory Prayer. American Heart Journal, 152(3), e25.
Galton, F. (1872). Statistical inquiries into the efficacy of prayer. The Fortnightly Review, 68, 125-135.
Harris, W., Gowda, M., Kolb, J., Strychacz, C., Vacek, J., Jones, P., Forker, A., O'Keefe, J., & McCallister, B. (1999). A randomized, controlled trial of the effects of remote, intercessory prayer on outcomes in patients admitted to the coronary care unit. Archives of Internal Medicine, 46(1), 1878, 2273-2278.
Hodge, D. R. (2007). A systematic review of the empirical literature on intercessory prayer. Research on Social Work Practice, 17, 174-187.
Howard, G., Hill, T., Maxwell, S, Baptista, T. M., Farias, M., Coelho, C., Coulter-Kern, M., Coulter-Kern, R. (2009). What’s wrong with research literatures? And how to make them right. Review of General Psychology, 13(2), 146-166.
Hunt, D. (2000). Remote, intercessory prayer reduced the frequency of adverse events in patients newly admitted to the coronary care unit. Evidence-based Cardiovascular Medicine, 4, 48.
Ikedo, F., Gangahar, D., Quader, M., & Smith, L. (2007). The effects of prayer, relaxation technique during general anesthesia on recovery outcomes following cardiac surgery. Complementary Therapies in Clinical Practice, 13, 85-94.
Joyce, C., Welldon, R. (1965). The objective efficacy of prayer: A double-blind clinical trial. Journal of Chronic Diseases, 18, 367-377.
Karis, R., & Karis, D. (2000). Intercessory Prayer. Archives of Internal Medicine, 160, 1870.
Krucoff, M., Crater, S., Gallup, D., Blankenship, J., Cuffe, M., Guameri, M., Krieger, R., Kshettry, V., Morris, K., Oz, M., Pichard, A., Sketch, M., Koenig, H., Mark, D., & Lee, K. (2005). Positive music, imagery, touch, and prayer as adjuncts to interventional cardiac care: The monitoring and actualisation of noetic trainings (MANTRA) II randomized study. The Lancet, 366, 211-217.
Krucoff, M., Crater, S., Green, C., Maas, A., Seskevich, J., Lane, J., Loeffler, K., Morris, K., Bashore, T., & Koenig, H. (2001). Integrative noetic therapies as adjuncts to percutaneous intervention during unstable coronary syndromes: Monitoring and actualization of noetic training (MANTRA) feasibility pilot. American Heart Journal, 142, 760-767.
Krucoff, M., Crater, S., Lee, K. (2005). From efficacy to safety concerns: A STEP forward or a step back for clinical research and intercessory prayer?: The study of therapeutic effects of intercessory prayer (STEP). American Heart Journal, 151(4), 762-764.
Leibovici, L. (2001). Effects of remote, retroactive intercessory prayer on outcomes in patients with bloodstream infection: randomised controlled trial. British Medical Journal, 323, 1450-1451.
Lesniak, K. T. (2006). The effect of intercessory prayer on wound healing in nonhuman primates. Alternative Therapies in Health and Medicine. 12(6), 42-48.
Lewis, C. A., Breslin, M. J., & Dein, S. (2008). Prayer and mental health: An introduction to this special issue of mental health, religion & culture. Mental Health, Religion & Culture, 11(1), 1-7.
Marks, L. D. (2008). Prayer and marital intervention: Asking for divine help... or professional trouble? Journal of Social and Clinical Psychology, 27(7), 678-685.
Masters, K. (2005). Research on the healing power of distant intercessory prayer: Disconnect between science and faith. Journal of Psychology and Theology, 33(4), 268-277.
Masters, K., Spielmans, G., & Goodson, J. (2006). Are there demonstrable effects of distant intercessory prayer? A meta-analytic review. Annals of Behavioral Medicine, 32(1), 21-26.
McCullough, M. E., & Larson, D. B. (1999). Prayer. In W. R. Miller (Ed.), Integrating spirituality into treatment: Resources for practitioners (pp. 85-110). Washington, D.C.: American Psychological Association.
Olver, I., & Whitford, H. (2009). Intercessory prayer improves spiritual wellbeing in a randomised controlled trial in patients with cancer. European Journal of Cancer Supplements, 7(2), 176.
Palmer, R., Katerndahl, D., & Morgan-Kidd, J. (2004). Lifetime a randomized trial of the effects of remote intercessory prayer: Interactions with personal beliefs on problem-specific outcomes and functional status. The Journal of Alternative and Complementary Medicine, 10(3), 438-448.
Price, J. (2000). Does prayer really set one apart? Archives of Internal Medicine, 160, 1873.
O'Mathuna, D. P. (2004). Commentary. Focus on Alternative and Complementary Therapies, 9(4), 315-316.
Sandweiss, D. (2000). P value out of control. Archives of Internal Medicine, 160, 1872.
Sicher, F., Targ, E., Moore II, D., Smith, H. (1998). A randomized double-blind study of the effect of distant healing in a population with advanced AIDS: Report of a small study scale. Western Journal of Medicine, 169(6), 356-363.
Sloan, R. P., & Bagiella, E. (2000). Data without a prayer. Archives of Internal Medicine, 160, 1870.
Van der Does, W. A randomized, controlled trial of prayer? Archives of Internal Medicine, 160, 1871.
Walker, S., Tonigan, J., Miller, W., Comer, S., & Kahlich, L. (2009). Intercessory prayer in the treatment of alcohol abuse and dependence: A pilot investigation. Alternative Therapies in Health and Medicine, 3(6), 79-86.