That's an excellent question. Here is a rough reply. Oftentimes, when we refer to some everyday phenomenon as "random", we mean that we are ignorant of the fundamental causes at work -- as in games of chance. However, according to modern physics, there are some fundamental phenomena involving the behavior of sub-atomic particles that are genuinely random. For example, if a radioactive atom existing now has a half-life of (let's say) 100 seconds, then there is a 50% chance that it will decay sometime during the next 100 seconds, and there is no feature that the atom has now (or that anything else has now) that determines whether the atom will decay or won't decay. It is an irreducibly random process. In other words, the atoms that ultimately do decay before 100 seconds have passed are no different now from the atoms that do not decay during that interval. There are no "hidden variables" to distinguish them. I should add that the reason we have for believing that these phenomena are genuinely random...

This is an excellent question. You are right: getting heads twenty times in a row is exactly as likely as, say, HTTHTHHTTTHHTHTHHTTH. However, 20 heads is much less likely than (say) 10 heads and 10 tails. There are many more twenty-flip combinations that yield 10 heads and 10 tails than twenty-flip combinations that yield 20 heads. There are more ways to get 10 H and 10 T than to get 20 H. So getting 20 heads is less likely (assuming the coin is fair) than getting 10 heads and 10 tails. Yet getting 20 heads is exactly as likely as getting a particular combination of 10 heads and 10 tails, such as HTTHTHHTTTHHTHTHHTTH. Other cases of "strangeness" are a bit more difficult to diagnose. For instance, suppose a lottery is run and ticket #1729 wins. This outcome is extremely unlikely if the lottery is fair (let's suppose there are 10,000 tickets), but extremely likely if the lottery was fixed for #1729. Does this mean that after this ticket is drawn, we should conclude that the lottery was...

Since you are obviously interested in probability and baseball, here's a fun question for you to think about. How can it happen that player A has a higher batting average than player B in the first half of the season, and A also has a higher batting average than B in the second half of the season, but B has a higher overall season batting average than A? (Yes, this can indeed happen. It is a form of "Simpson's Paradox.)

A fair coin may land "heads" over and over again, as we all know. As the number of tosses increases without bound, the likelihood of the coin's landing exclusively on "heads" becomes arbitrarily small. So in the limit, the likelihood of its landing "heads" over and over (or even 75% of the time) is zero. Nevertheless, it is possible (where a "fair" coin, by definition, has a 50% chance of landing heads and a 50% chance of landing tails).

Someone who plays the lottery more than once stands a greater chance of winning than someone who plays it on only one occasion. Compare: The chance of rolling "six" once with a fair die is greater if you roll the die twice than if you roll it once. The chance of your rolling "six" on one toss is (naturally!) 1/6. Your chance of rolling "six" once in two tosses is 1 minus your chance of rolling 1-5 on the first toss and 1-5 on the second toss, i.e., 1 - (5/6)(5/6), which equals 11/36 -- which is 5/36 more than 1/6. However, you said that given enough trials of the lottery (having a finite number of tickets), every ticket will eventually win. That's not true. It's like saying that if you are tossing a fair coin, then it is guaranteed that a head will eventually appear. That's not true. You *could* get all tails. As the number of throws increases, the chance of getting all tails diminishes, and with an infinite number of throws, the chance of getting all tails is zero. But that does not mean that it...

That is not one of the definitions of "randomness" with which I am familiar. Rather than consider how to define "randomness", let's consider the idea of every possible outcome having an equal chance of occurring. Suppose I ask you to select a prime number "at random". What would it be for every possible outcome to have an equal chance of being selected? One way to understand this would be for every prime number to have the same chance of being selected. But there are an infinite number of prime numbers. So the only way for each prime number to have the same chance of being selected is for each to have zero chance of being selected. However, it would then seem reasonable to conclude that the chance of your selecting *any* prime number at all is the chance of your selecting the first prime number (which is 1) plus the chance of your selecting the second prime number (2) plus the chance of your selecting the third prime number (3) plus the chance of your selecting the fourth prime number ...