playing dice for the last banana.

玩骰子決定誰拿走最後一根香蕉

You've agreed on these rules:

你們都同意這些規則

You'll roll two dice,

你們將擲兩個骰子

and if the biggest number is one, two, three or four,

如果最大的數字是 1到4

player one wins.

第一名玩家獲勝

If the biggest number is five or six, player two wins.

如果最大的數定提5 或6 第二名玩家獲勝

Let's try twice more.

咱們多試兩次

Here, player one wins,

這裡，第一名玩家獲勝

and here it's player two.

而這裡，則是第二名玩家獲勝

So who do you want to be?

那你想當哪一個玩家呢？

At first glance, it may seem like player one has the advantage

第一印象， 似乎是第一名玩家佔優勢

since she'll win if any one of four numbers is the highest,

因為如果四個數字的任一個為最大，她將獲勝

but actually,

但，事實上

player two has an approximately 56% chance of winning each match.

第二名玩家有56%的機率會獲勝

One way to see that is to list all the possible combinations you could get

用一種方法，來看看你所能得到的所有組合

by rolling two dice,

從擲兩個骰子的活動中

and then count up the ones that each player wins.

然後計算每個玩家獲勝的次數

These are the possibilities for the yellow die.

這些是黃骰子可能的結果

These are the possibilities for the blue die.

這些是藍骰子可能的結果

Each cell in the chart shows a possible combination when you roll both dice.

表中的每個格子 代表擲兩個骰子的可能組合

If you roll a four and then a five,

若你擲出一個4 和一個5

we'll mark a player two victory in this cell.

我們就註記第二名玩家 在這格獲勝

A three and a one gives player one a victory here.

一個3 和一個1 代表第一名玩家獲勝

There are 36 possible combinations,

總共有36種組合

each with exactly the same chance of happening.

每一種發生的機率都一樣

Mathematicians call these equiprobable events.

在數學中稱為 相等機率事件

Now we can see why the first glance was wrong.

現在我們可以看到 為何第一印象是錯的

Even though player one has four winning numbers,

即使第一名玩家有4個獲勝數字

and player two only has two,

而第二名玩家只有2個獲勝數字

the chance of each number being the greatest is not the same.

每個數字成為最大值的機率都一樣

There is only a one in 36 chance that one will be the highest number.

在36種組合中 只有一種組合的最大值是1

But there's an 11 in 36 chance that six will be the highest.

但在36種組合中 有11種組合的最大值是6

So if any of these combinations are rolled,

所以，若擲出這些組合的任一種

player one will win.

第一名玩家獲勝

And if any of these combinations are rolled,

若擲出這些組合的任一種

player two will win.

第二名玩家獲勝

Out of the 36 possible combinations,

在36種可能的組合中

16 give the victory to player one, and 20 give player two the win.

16種由第一名玩家獲勝 20種由第二名玩家獲勝

You could think about it this way, too.

你也可以用這個方法思考

The only way player one can win

第一名玩家僅會在這樣的時候獲勝

is if both dice show a one, two, three or four.

當兩個骰子都擲出1到4

A five or six would mean a win for player two.

任一個5 或 6 代表第二名玩家獲勝

The chance of one die showing one, two, three or four is four out of six.

一個骰子出現1到4的機率是 六分之四 (4/6)

The result of each die roll is independent from the other.

個別骰子都是獨立事件

And you can calculate the joint probability of independent events

你可以計算這些獨立事件的聯合機率

by multiplying their probabilities.

經由 他們的機率的相乘

So the chance of getting a one, two, three or four on both dice

所以 兩個骰子都擲出 1，2，3，或4的機率是

is 4/6 times 4/6, or 16/36.

六分之四 乘以 六分之四 (4/6 * 4/6)

Because someone has to win,

因為總有人要獲勝

the chance of player two winning is 36/36 minus 16/36,

第二名玩家獲勝的機率是 三十六分之三十六 減去 三十六分之十六 (36/36 - 16/36)

or 20/36.

即 三十六分之二十 (20/36)

Those are the exact same probabilities we got by making our table.

這跟表格算出來的機率正好相同

But this doesn't mean that player two will win,

但這不代表第二名玩家會贏

or even that if you played 36 games as player two, you'd win 20 of them.

也不代表第二名玩家在36次比賽中會贏20次

That's why events like dice rolling are called random.

這就是為何擲骰子被稱為 隨機事件

Even though you can calculate the theoretical probability

即使你可以算出理論上的機率值

of each outcome,

每個結果的機率值

you might not get the expected results if you examine just a few events.

你可能得不到預期的結果 如果你只試驗了幾次的話

But if you repeat those random events many, many, many times,

但你若重複隨機事件 很多、更多、超多次的話

the frequency of a specific outcome, like a player two win,

特定結果出現的頻率 例如第二名玩家獲勝

will approach its theoretical probability,

將會接近理論上的機率值

that value we got by writing down all the possibilities

那就是我們列出所有可能組合

and counting up the ones for each outcome.

再把各種結果的機率值加總所得到的值

So, if you sat on that desert island playing dice forever,

所以，如果你在荒島上 持續不斷的擲骰子

player two would eventually win 56% of the games,

第二名玩家會贏得百分之五十六 (56%) 的比賽

and player one would win 44%.

而第一名玩家會贏得百分之四十四 (44%) 的比賽

But by then, of course, the banana would be long gone.

但屆時，當然，香蕉早就消失了