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Actually that makes my point stronger. You stated that people only believe based on where they're born. But that's not the case.
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No it only works if converts only go towards Christianity because that would be a decent indicator that it has an inherent truth that causes people to shift beliefs. That is not what happens so what you are left with are people that find themselves at a time and in a place that is conducive for them to change beliefs.
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We're not talking about only Christianity. You brought up Islam too. Your assertion that people's beliefs are only based on their origin, is erroneous. People change beliefs all the time in many different directions. People believe what they have found to be true, regardless of how they were exposed to it.