stevieb1,
I was just praying that somone would ask that question.
You can't really argue with it. It's the ideal fantasy for the door-to-door work. If the stories never really happened, they would be created. But rationally, we could assume that anyone who is truly desperate will have felt that way over an extended period of time, and will likely have a better memory of the prayer that was "answered" than the 40 times they prayed the same prayer and no one came to the door.
Also, such people will be more likely to be home on a weekend than out playing with friends. They will answer the door for anyone just to try to break a depression or monotony or loneliness. Most of the JW door-to-door converts were temporarily "without a life" or "without an activity/religion" and what we had to offer must have sounded better than anything they had in their current condition. These same people wouldn't have given us JWs the time of day the previous month, but whatever got them down, got us a foot in the door. Timing is everything!
I assume that Mormons and others also use the same experience so I did a quick google search on the phrase: [end of my rope "knock at the door"]
Ignoring a few sites with themes or poems about Jesus knocking at your door when you're at the end of your rope, and a few other sites about some kind of bondage, I found two interesting ones that prove that these stories would be invented even if they never happened:
http://www.geocities.com/Athens/Crete/6111/pneumatikos/dragnet.htm
Soul Patrol, a Christian youth skit about door-to-door evangelism ...
Sunday: After hitting every house for a mile and a half. My tootsies are starting to bark like dogs singing the blues. Mundy's shoes are about ready for a retread. We decide to hit one more house before breaking for coffee and donuts. We are tired.
Miss Guide: Well, yes. You mean, you're not my new spiritual advisor, and Rafka, "Channel to the Stars"? I was expecting them.
Sunday:
No, ma'am. We're the Soul Patrol. We'd like to talk to you about Jesus Christ, the only real way to eternal life. [edited out a bit...]
Mundy: Yeah. Jesus shed His blood for your sins. If you believe in Him, no matter what you've been, no matter what you've done, Jesus will forgive you. You can start over again with a clean slate. But this time, He'll be there to help you do better.
Needy:
Gee, that sounds awful good. I gotta tell you guys, things have been so bad, I was making plans to end it all. You know, suicide.
http://www.homeandholidays.com/rs/pages/lead-vt-skit2.html (seemed to be unavailable, so I used the Google cached version)
Here's it's conclusion after a similar skit about Sisters on errands of mercy:
Narrator: Here's a common case. The visiting teachers just assume they're not important to the people they visit. And the people they visit sometimes forget to mention how much they love to see them. These sisters provided real help to the women they went to visit. Some of it was physical labor, but it was all a labor of love. Once they got up and got motivated, they felt really good about the whole thing. The spirit prompted these sisters to go and do their visiting teaching and at first, they ignored it. We want you to recognize your worth. Visiting teaching IS important. It's a divine program and this was a small example of it in action. The scenes portrayed here were based on actual visiting teaching experiences. There was a woman laying on the floor, begging for her Heavenly Father to send her angels on an errand of mercy. It just so happened that the Father sent the sisters. And believe it or not, a piece of chocolate cake, delivered by a sister, saved a woman's life. The woman begging for a simple piece of chocolate cake had been in her bedroom contemplating suicide at the moment her visiting teachers came in. The ONLY thing that would save her from doing such a sad thing was a single piece of cake. She was very specific in her prayers. And there you have it...visiting teachers answer prayers every day. Whose prayers have you answered? Sisters, you are angels on errands of mercy. Recognize the divine origin of this program and take part in it with a heart full of song. Make the time to become friends with the women you are called to serve.
Hope it helps,
Gamaliel