# In the probability thread, multiple philosophers mention examples of zero-probability events that aren't necessarily "impossible" (like flipping an infinite number of heads in a row). Arriving at a probability of zero in these instances relies on saying that 1/infinity = 0. But this math seems misleading. Don't mathematicians rely on more precise language to avoid this paradoxical result, by saying that "the limit of 1/x as x approaches infinity = 0," rather than simply "1/x = 0"? I feel like there must be some way to distinguish (supposedly) zero-probability events that are actually possible and zero-probability events that are impossible. Thanks!

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...

# A friend posed a problem that according to him reveals an inconsistency in mathematics. There are two envelopes with money in them, and you are given one envelope. One envelope has twice the amount of money as the other, but you don't know which one is which. The question is, if you are trying to maximize your money, after you are given your envelope, should you switch to the other envelope if given the chance? One analysis is: let a denote the smaller amount. Either you have a or 2a in your envelope, and you would switch to 2a or a, respectively, and since these have the same chance of happening before and after, you don't improve and it doesn't matter if you switch. The other analysis is: let x denote the value in your envelope. The other envelope has either twice what is in yours or it has half of what is in yours. Each of these has probability of .5, so .5(2x) + .5(.5x) = 1.2x, which is greater than the x that you started with, so you do improve and should switch. Is there something wrong with...

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? ...

# Is an event which has zero probability of occurring but which is nonetheless conceivably possible rightly termed "impossible"? For instance, is it "impossible" that I could be the EXACT same height as another person? I take it that the chance of this is zero in that there are infinitely many heights I could be (6 ft, 6.01 ft, 6.001 ft, 6.0001 ft, etc.) but only one which could match that of a given other person exactly; at the same time, I have no problem at all imagining a world in which I really am exactly as tall as this other.

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...

# My girlfriend and I had a discussion about probability as it relates to a weekly lottery draw. She argued that the probablity of winning remains the same from draw to draw, and because of this anyone who plays the lottery more than once stands no greater chance of winning than someone who only plays it on one occasion. Against this, I argued that because any lottery operates with a finite series of numbers, given enough draws all possible combinations will eventually have appeared at least once, and as such someone who plays more than once stands a greater chance of winning. I also claimed that the probability relating to each draw is different from that which relates to a succession of draws (again because of the finite series of numbers). Which of us is right?

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...

# Can there be an event that is entirely random?

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...