To answer this question, it may be helpful to say something about the mathematical formalism usually used in probability theory. The first step in applying probability theory to study some random process is to identify the set of all possible outcomes of the process, which is called the sample space . For example, in the case of an infinite sequence of coin flips, the sample space is the set of all infinite sequences of H's and T's (representing heads and tails). Probabilities are assigned to events , which are represented by subsets of the sample space. For example, in the case of an infinite sequence of coin flips, the set of all HT-sequences that start with H represents the event that the first coin flip was a heads, and (assuming the coin is fair) this event would have probability 1/2. The set of sequences that start with HT is a subset of the first one, and it represents the event that the first flip was heads and the second tails; it has probability 1/4. Now, consider some infinite HT...

I'd like to add a little bit to what Thomas has said. Probability problems can be tricky because the answers sometimes depend on small details about exactly what procedure was followed. For example, the problem says that "you are given one envelope." Who gave you the envelope? Did the person who gave you the envelope know which envelope was which? Was he a very stingy person, who might have been more likely to give you the envelope with the smaller amount of money? If so, then the probability that you have the smaller amount might not be 1/2. But that is clearly not the intent of the problem, so let us assume that the person who gave you the envelope flipped a coin to decide which envelope to give you. Then, as Thomas says, the probability is 1/2 that you have the small amount and 1/2 that you have the large amount. Suppose that you open your envelope and find $100 in it. You now know that the other envelope contains either $50 or $200. Do these two outcomes still have probability 1/2 each? ...

I agree that there's nothing paradoxical here; surprising, perhaps, but not paradoxical. The only kind of additivity that is usually assumed in probability theory is countable additivity, and there's no violation of that here. But you do have uncountably many non-overlapping outcomes, each with probability zero, such that the probability of at least one of those outcomes happening is one. So uncountable additivity doesn't work. I would agree that an outcome with probability zero need not be impossible. Consider, for example, flipping a coin infinitely many times. Each infinite sequence of heads and tails has probability zero of occurring, but one of them has to occur, so it wouldn't make sense to say that they're all impossible. (Notice that there are uncountably many possible sequences of heads and tails.) But of course this is not a realistic experiment--no one can actually flip a coin infinitely many times. The original example proposed also seems unrealistic to me--according to...

Marc is right that if you play the lottery more than once, your chance of winning at least once is higher than if you only play once. However, there is another possible interpretation of your question. Supppose you have played the lottery many times and lost every time. Is your chance of winning the next time higher than if you hadn't played before? Some people think that your chance of winning is now higher because the lottery "owes" you a win. But this is wrong; your chance of winning the next time is exactly the same as if you hadn't played before. This does not contradict Marc's answer. This is because probabilities depend on the precise situation, and when the situation changes, the probabilities can change. Let's consider again Marc's case of rolling a die twice. Before you start rolling the die, the probability of getting at least one 6 is, as Marc says, 11/36. But now suppose you do your first roll of the die, and you don't get a 6. Now the only way of getting a 6 on one of the two...

This is a very difficult question, for two reasons: 1. It is difficult to say exactly what "random" means. 2. There are unresolved questions in the foundations of quantum mechanics that are relevant to your question. Consider, for example, flipping a fair coin. This seems random, in the sense that we don't seem to be able to predict the outcome. Half the time the coin comes up heads and half the time it's tails, and we don't know which it's going to be until it lands. But in another sense, it doesn't seem random at all: If you knew the speed at which the coin was spinning, its exact position above the table, the air currents in the room, etc., then the laws of physics should allow you to predict how it will land. If you think of randomness as being about our lack of knowledge of how things are going to turn out, then the coin flip seems random. If you think of randomness as being about some sort of indeterminacy in the world, independent of our knowledge, then the coin flip doesn't seem...