WEBVTT

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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.