There’s several problems here. First of all, I almost always disregard people who make claims like yours (2^750,000,000:1) about the real world on account that they are almost always wrong or misleading. Specifically, while that sort of odds can exist it almost never exists in a way that would win someone points in a conversation. Such claims are often lies, miscalculations, or misleading. While your friend was equally wrong in considering the odds of such being compatible with “coincidence” as it is used in mathematics, how exactly do you expect to calculate the other type of meaning (maybe by “coincidence” God decided to use mostly the same DNA in humans and chimps)? Is it really fair to say that Probability(chimps and humans share 95-98% DNA given that God exists) = 2^750,000,000:1 or anything remotely close?
Is it really fair to claim that your friend was saying, “Well since you claim 2^750,000,000:1 odds instead of zero, I’m going to go with those odds” as opposed to “Even if you said the odds were zero, I wouldn’t believe you because there’s a chance you’re wrong”? There’s plenty of examples of people being certain of things, yet being wrong—even when they use math.
Is it rational to assign even claimed 10^9:1 odds anything remotely close to actual 10^9:1 odds? Seldom, I should say. I’d give such a claim a probability of something like 0.1%-75% of being flat out wrong, based on the difficulty of the problem, the contentiousness of the problem, my respect for the ability and integrity of the person making the claim, and whether the claim agrees or disagrees with things I know or think I know. Now ideally, if I have the time and capability, I would try doing some of those calculations myself and think a while as to whether those are even the correct calculations, but often claims won’t be worth that level of effort.
There’s several problems here. First of all, I almost always disregard people who make claims like yours (2^750,000,000:1) about the real world on account that they are almost always wrong or misleading. Specifically, while that sort of odds can exist it almost never exists in a way that would win someone points in a conversation. Such claims are often lies, miscalculations, or misleading. While your friend was equally wrong in considering the odds of such being compatible with “coincidence” as it is used in mathematics, how exactly do you expect to calculate the other type of meaning (maybe by “coincidence” God decided to use mostly the same DNA in humans and chimps)? Is it really fair to say that Probability(chimps and humans share 95-98% DNA given that God exists) = 2^750,000,000:1 or anything remotely close?
Is it really fair to claim that your friend was saying, “Well since you claim 2^750,000,000:1 odds instead of zero, I’m going to go with those odds” as opposed to “Even if you said the odds were zero, I wouldn’t believe you because there’s a chance you’re wrong”? There’s plenty of examples of people being certain of things, yet being wrong—even when they use math.
Is it rational to assign even claimed 10^9:1 odds anything remotely close to actual 10^9:1 odds? Seldom, I should say. I’d give such a claim a probability of something like 0.1%-75% of being flat out wrong, based on the difficulty of the problem, the contentiousness of the problem, my respect for the ability and integrity of the person making the claim, and whether the claim agrees or disagrees with things I know or think I know. Now ideally, if I have the time and capability, I would try doing some of those calculations myself and think a while as to whether those are even the correct calculations, but often claims won’t be worth that level of effort.
There’s a post by Yvain which addresses more or less this issue.