Probability Puzzle - Do angels exist?

#1
Diego, a farmer from a small town in Peru, has a dream one night in which an angel appears and informs him that he has led a good life and she will reward him by telling him the winning numbers of the state lottery for the very next evening. As soon as he wakes he scribbles them down before telling his brother excitedly. They then rush to buy a ticket with the angel’s recommendations – the first one they have ever bought.

That evening all the numbers come up, and the brothers win the jackpot – a one in ten million chance! They are now rich beyond their wildest dreams.
I read about this extraordinary ‘angel vision’ in the newspapers; you really couldn’t miss it as it was reported everywhere. I’m so intrigued on the subject of any supernatural that caught a plane from my home in London all the way to the small town in Peru to track down the brothers. Hopefully I would uncover more information on this mysterious event.

After much searching, I located them in a bar on the edge of town and soon got chatting. After a few drinks, I asked “so do you believe in angels?”. Diego said he must; he explains the chances of a person having a dream on any night in their lifetime in which they can remember any six numbers is at least, say, one in twenty thousand. He didn’t know anyone who had experienced this in his small town at least. Therefore the probability of having a dream and recalling any six numbers and then winning the Peruvian lottery from a single ticket is around one in 200 billion.

I remarked that “maybe Angels do exist, at least in dreams!”. But his brother said that although he also believed an angel visited his brother, I shouldn’t. He reasoned that as I was an avid reader of world news, plus someone with an interest in the supernatural, the chances of me reading about someone who won a state lottery immediately after a premonition were reasonable. Even in the past ten years there would have been approximately 100,000 state lottery draws across the world. If there were on average five million people purchasing tickets in each, that would be 500 billion potential winners – or 500 billion opportunities for someone to have a dream about the lottery numbers the evening prior to a draw.

I thought his reasoning was sound for far. Yet he went further: You would expect around 25 million ‘lottery dreams’ across the media-covered world in ten years, enough to make it more than likely that one ticket purchased on basis of the dream would win a big lottery jackpot, such as Peru’s. Any such story would have a high chance of being reported internationally. The chances of you reading such an article are high, and the chances of you personally travelling to meet the lucky winners are well, perhaps near 100%.

“So I shouldn’t believe in Angels, but you should?” I asked. “Yes” they both nodded.
I knew this is an absurd line of reasoning so dismissed it and went to the bar to get another drink. The barman, on overhearing our conversation, said he would be happy to give me a beer ‘on the house’, but only if I told him whether angels were real. What should I tell him?