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You and a fellow castaway
are stranded on a desert island
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playing dice for the last banana.
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You've agreed on these rules:
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You'll roll two dice,
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and if the biggest number
is one, two, three or four,
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player one wins.
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If the biggest number is five or six,
player two wins.
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Let's try twice more.
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Here, player one wins,
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and here it's player two.
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So who do you want to be?
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At first glance, it may seem
like player one has the advantage
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since she'll win if any one
of four numbers is the highest,
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but actually,
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player two has an approximately
56% chance of winning each match.
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One way to see that is to list all
the possible combinations you could get
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by rolling two dice,
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and then count up
the ones that each player wins.
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These are the possibilities
for the yellow die.
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These are the possibilities
for the blue die.
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Each cell in the chart shows a possible
combination when you roll both dice.
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If you roll a four and then a five,
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we'll mark a player two
victory in this cell.
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A three and a one gives
player one a victory here.
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There are 36 possible combinations,
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each with exactly the same
chance of happening.
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Mathematicians call these
equiprobable events.
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Now we can see why
the first glance was wrong.
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Even though player one
has four winning numbers,
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and player two only has two,
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the chance of each number
being the greatest is not the same.
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There is only a one in 36 chance
that one will be the highest number.
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But there's an 11 in 36 chance
that six will be the highest.
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So if any of these
combinations are rolled,
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player one will win.
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And if any of these
combinations are rolled,
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player two will win.
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Out of the 36 possible combinations,
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16 give the victory to player one,
and 20 give player two the win.
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You could think about it this way, too.
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The only way player one can win
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is if both dice show
a one, two, three or four.
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A five or six would mean
a win for player two.
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The chance of one die showing one, two,
three or four is four out of six.
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The result of each die roll
is independent from the other.
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And you can calculate the joint
probability of independent events
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by multiplying their probabilities.
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So the chance of getting a one, two,
three or four on both dice
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is 4/6 times 4/6, or 16/36.
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Because someone has to win,
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the chance of player two winning
is 36/36 minus 16/36,
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or 20/36.
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Those are the exact same probabilities
we got by making our table.
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But this doesn't mean
that player two will win,
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or even that if you played 36 games
as player two, you'd win 20 of them.
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That's why events like dice rolling
are called random.
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Even though you can calculate
the theoretical probability
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of each outcome,
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you might not get the expected results
if you examine just a few events.
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But if you repeat those random events
many, many, many times,
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the frequency of a specific outcome,
like a player two win,
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will approach its theoretical probability,
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that value we got by writing down
all the possibilities
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and counting up the ones for each outcome.
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So, if you sat on that desert island
playing dice forever,
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player two would eventually
win 56% of the games,
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and player one would win 44%.
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But by then, of course, the banana
would be long gone.