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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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