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• You and a fellow castaway are stranded on a desert island

• playing dice for the last banana.

• You've agreed on these rules:

• You'll roll two dice,

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

• player one wins.

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

• Let's try twice more.

• Here, player one wins,

• and here it's player two.

• So who do you want to be?

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

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

• but actually,

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

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

• by rolling two dice,

• and then count up the ones that each player wins.

• These are the possibilities for the yellow die.

• These are the possibilities for the blue die.

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

• If you roll a four and then a five,

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

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

• There are 36 possible combinations,

• each with exactly the same chance of happening.

• Mathematicians call these equiprobable events.

• Now we can see why the first glance was wrong.

• Even though player one has four winning numbers,

• and player two only has two,

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

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

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

• So if any of these combinations are rolled,

• player one will win.

• And if any of these combinations are rolled,

• player two will win.

• Out of the 36 possible combinations,

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

• You could think about it this way, too.

• The only way player one can win

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

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

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

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

• And you can calculate the joint probability of independent events

• by multiplying their probabilities.

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

• is 4/6 times 4/6, or 16/36.

• Because someone has to win,

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

• or 20/36.

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

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

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

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

• Even though you can calculate the theoretical probability

• of each outcome,

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

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

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

• will approach its theoretical probability,

• that value we got by writing down all the possibilities

• and counting up the ones for each outcome.

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

• player two would eventually win 56% of the games,

• and player one would win 44%.

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

You and a fellow castaway are stranded on a desert island

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B1 US TED-Ed player chance probability roll outcome

# 【TED-Ed】The last banana: A thought experiment in probability - Leonardo Barichello

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稲葉白兎 posted on 2015/03/09
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