A lot of people don't
believe that God answers prayers. I will show you why we can be reasonably sure
He does. God answering prayers is basically this; a Christian prays for x, x
happens, therefore x happened because God answered the prayer. That happens a
lot, but this apparent pattern is complicated by two common exceptions: times
when a Christian prays for x, and x does not occur; and times when x occurs
without anyone praying for it at all.
It is easy to understand why both exceptions occur, though; when a Christian prays for x, and x does not occur, we ought to conclude that x was the wrong prayer or not aligned with God's will. And x occurring without anyone praying for it at all does not tell us anything much about whether God answers prayers, any more than a fair coin landing heads tells us whether coins in general are biased.
A more appropriate way to affirm that God answers prayers is to look at what is being prayed for, and ask what the likelihood of the event is anyway without the prayer. For example, suppose Jack has a deck of cards, and prays that, after shuffling them, he will draw the king of diamonds. Since there are 52 cards in the deck, there is a 1 in 52 probability of this happening by chance alone. If 52 independent people were each to pray that they would draw the king of diamonds from their shuffled deck, probability suggests that, on average, one of them would succeed and might therefore conclude that their prayer had been answered.
What we are therefore looking for, in order to justify belief in answered prayer, is to consider events that occur where there is an astronomically low probability that it would happen by chance, but which are specifically and unambiguously prayed for in advance. This requirement precludes cases in which low-probability events are later interpreted as answers to prayer simply because they happened to occur, and excludes low-probability events that do in fact occur by chance, but lack any prior, specific prayer corresponding to them.
At this point, the sheer weight of Christian testimony should be ample evidence that God answers prayers - the kind of prayers which significantly undermine objections such as 'it is only anecdotal testimony' or 'there is no medical verification'. I have been instantly healed a few times immediately after prayer - from a pulled muscle in my leg, and from a chronic tooth pain - and I have witnessed a blind person given their sight back immediately after prayer, a severely deformed leg twist around, grow and be restored immediately after prayer, and a lady crippled and confined to a wheelchair all her adult life stand up and walk immediately after prayer. And they are merely a few experiential drops in a sea of miraculous testimony amassed worldwide.
The likelihood of any one of those events happening by chance is astronomically low, so the likelihood of any one of those events happening by chance immediately after prayer is even lower (for obvious reasons). Therefore, the most likely explanation is that these immediate healings were answers to prayer - especially when these events are considered cumulatively. If the likelihood of an instant healing is almost zero and it happens straight after prayer, the best explanation is the prayer caused it to happen. If the same thing repeats five times in just my experience, the best explanation being answered prayer becomes cumulatively stronger. When this pattern is multiplied across the experiences of Christians more broadly, answers to prayer become so much more plausible that it would be difficult to dismiss without adopting an unjustified scepticism.
