Interesting Paradoxes

Wanna share some of the most interesting paradoxes I have read. Full details can be found at wikipedia.

1) Out of merely 57 randomly chosen people, there is 99% probability that there is a pair or more with the same birthday.
Some reasoning: It only states that a pair or more have the same birthday. It means any pair and take note that there are a lot of pairs possible. It does not state such a probability on a particular person. So, if you are in a group of 57, having a person with same birthday as you is far less than 99%.

2) Suppose you have seen a dice fell on six 10 times. If the dice is fair, is it less likely for the next roll to be six?
NO. Landing on six for 11th time is exactly as likely as landing on any other numbers.
Some reasoning: This is a consequence of independent events. We know that and it is true that getting six for 11 consecutive times is very unlikely (1/6)^11. But, this case is different. The dice has already landed on six for ten times. So, in a way, since the unlikely case has already happened, the next roll has no unlikeness anymore.

3) A tortoise and Achilles run a race (of course, Achilles is much faster than the tortoise). The tortoise gets a head start. We reason as follows that Achilles will never overtake the tortoise. Since the tortoise is ahead of Achilles, Achilles needs to get to the position of tortoise before overtaking it. Achilles will need some time to get to the tortoise position. But, when Achilles reaches the tortoise position, since some time has lapsed, the tortoise is now in a new position. The same thing happens again when Achilles tries to reach that new position. This keeps on happening and Achilles will not overtake tortoise.
Some reasoning: This obviously is incorrect. Achilles will overtake the tortoise at one point. There are a lot of explanation to this problem from many viewpoints here and here. But, one simple reasoning I can think of is that for increasingly smaller distances Achilles needs to cover, the corresponding time taken will also become increasingly shorter. So, there will be an infinitely small distance he needs to cover and the time taken for that distance would also be infinitely short. An infinitely short amount of time means instantly. So, he can cover that distance instantly and can catch up the tortoise.