A Coin Tossing Game

Here's the rule: With a fair coin, Helen gets a point when it flips head and Tim gets a point when it flips tail. Helen wins when she becomes 3 points ahead. Tim wins when he becomes 10 points ahead. How much more likely is Helen to win? And if we use a weighted coin, what should the weight be to make this into a 50/50 fair game?

0.999.... = 1

I wrote this in respond to someone's question whether 0.9999.... = 1 in an HK forum. I liked it so much that I am pasting it here:

嚴格來說，小數并不代表實數、而是代表運算嘅一種簡寫（對人類來說比較方便）方法。

當你寫一個（10進制）小數嘅時候，例如0.33，這4嘅符號（0 . 3 3）代表嘅係以下運算嘅結果 它是一個實數（因為實數有限個基本運算下閉合）。而我們把這運算簡寫成0.33。

當你寫一個無限小數嘅時候，例如0.3333....代表嘅係以下无限級數嘅運算結果

這時候你就必須先問清楚咩係『無限個運算嘅結果』因為你不可能真係去做無限個運算。很自然你會回答：無限個運算我『識』做，因為我學過极限和收敛等概念！
跟住你必須問：咁呢個級數是否收敛到一個實數？如果不能收敛到一個實數、當然不能代表任何實數。好彩答案是肯定的：實數有完备性、以上數列是柯西序列、所以它收敛於一個實數（但這些都不是小學能輕易證明的事情）。和1/3結果一樣嘅實數。

所以說0.9999.... = 1只不過係表示 這個級數嘅收敛极限和幾個運算結果相等。至於點證明，自己Google。

Series are fun.

We learned in grade school that order in which one adds a bunch of numbers doesn't matter. So . This is not necessarily true when you are adding infinitely many numbers. Consider this series: and this one: The question is: are they equal?