TTA:
Well, I don't think you could have stated the numbers in a more confusing way than you did; do you mean you have 2 brothers and 2 sisters? or just 2 siblings total? I'll assume the latter for arguments' sake. (This weakens my case - if you have more siblings, the 3 birthdays become more likely.)
So you have, in total, 3 parents + 4 siblings + 4 cousins + you = 12 people in the group where it would be 'odd' to have 3 peoples' birthdays the same. And a total of 365 days that birthday could be.
A word about notation; in stats it's most simple to express probabilities as fractions between 1 and 0, rather than percentages. By this (linear) scale, 1 is an absolute certainty, and 0 and absolute impossibility (neither of which actually exist in reality). If you still feel the need to convert a number into a percentage, just multiply by 100; thus odds of .5 are the same as 50%.
Now for the math.
Let's start with a group of 10 people all with different birthdays, and find the odds that the next two people would share one of those birthdays. This gives 10/365 that one will share the birthday, and 10/365 * 1/365 = .00007506 that the third will share the same. So far not terribly good odds - still better than the 6/49 lottery, though. But, there's more. That group of ten we started with has odds of already sharing a birthday. The odds that 10 randomly picked birthdays will all be different is 365/365 * 364/365 * 363/365 * ... * 356/365 = .8830 , which means the odds they'll already have two dates shared is .1170 and then, factoring in the two people we left out, the odds of three same birthdays is .1170 * 2/365 = .0006408 . Combining this and the previous answer, we get .0006408 + .00007506 * .8830 = .0007071 which is (1/.0007071 = 1414) roughly 1 in 1500 . Pretty decent odds for what you consider a freak occurance.
To guarantee the same odds when playing a 6/49 lottery, you would have to buy 49C6 * .0007071 = almost 10,000 tickets. In the world there are probably about 6,000,000,000 * .0007071 = 4.24 million families with the same 'coincidence' - though it can hardly be called a coincidence in the presence of such a large number of occurences. If you know about 500 people (from different families), odds are roughly (1 - .0007071)^500 = 30% that one of them will have witnessed the same 'coincidence'.
If anyone else reading this knows a bit of stats, I'd appreciate a double-check.
As for the teacup thing, well, yeah, stats says that it's extremely unlikely, almost infinitesimally likely, that a teacup would break the same way twice. The odds of each crack happening in the exact same place are overwhelmingly small. So in effect, it says that it's 'impossible' - to the degree that anything can be impossible.
Shadow:
It's a tough realm, applying stats to real life, because they're by nature not the easy answers we're looking for. Just because something has a 99.99999% chance of being a beer can doesn't mean it actually is one. If you want to look at it from that point of view, all reality is subjective because there's an infinitesimally small chance that everything one percieves was actually a fluke accident of the light, and never really happened. Even events in the past are not 100% certain; Heisenberg's uncertainty principle. Quantum experiments have been done in which the actual physical reality of what happened to a photon is changed, after the fact, by measurement.
