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This presentation looks at the dice used in the 4000 year old Royal Game of Ur discovered in the 1920's by C. Leonard Woolley in the ruins of the ancient city of Ur in Mesopotamia
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A look at chance and probabilityin the
Royal Game of Ur
In 1922 C. Leonard Woolleydiscovered a 4000 year old gamein a tomb in Ancient Ur.
The Royal Game of Urhad 5 playing piecesAnd 4 dice per player.
© Trustees of the British Museum
Each die was a 4-sided equilateral pyramid. Two of the 4 corners where marked white. This meant that when a die was rolled, there were 2 chances in 4 of getting a white corner.
=
So the probability of rolling 1 (if a white corner counts as a 1) is
=
If a player has 4 pyramid dice to roll, then a 0, 1, 2, 3 or 4 could be rolled.
Is there an EQUAL chance of these scores happening?
We will draw a tree diagram to help us answer this question.
A white corner will be represented by a W and a black corner with a B. Every white at the top after a throw counts as 1.
W
B
This would count as 0.
In the tree diagram there will be 4 events matching the results of the 4 dice thrown.
W
B
First dice =
= 12
12
So the chance of rolling a 1 (a white tip)is 1 chance in 2.
W
W
B
B
W
B
1st dice 2nd dice
12
12
12
12
12
12
When the 2nd pyramidis rolled, once again thereis a 1 chance in 2 of gettingAnother white tip.
The chance of getting twowhite tips (rolling a 2) is found by multiplying alongthe branch of the treediagram.
This means that the chanceof rolling a 2 is
This is 1 chance in 4.
×
12×12=𝟏𝟒
W
W
B
B
W
B
1st dice 2nd dice
12
12
12
12
12
12
The same can be done for each branch of the tree.
×
Probability of rolling a 2 is
Probability of rolling a 1 is
Probability of rolling a 1 is
Probability of rolling a 0 is
W
W
B
B
W
B
1st dice 2nd dice
12
12
12
12
12
12
The same can be done for each branch of the tree.
×
Probability of rolling a 2 is
Probability of rolling a 1 is
Probability of rolling a 1 is
Probability of rolling a 0 is
Since there are two
possibilities of rolling a 1, we add the fractions.
This gives a or chance
of rolling a 1.
The combined probability of rolling a 1 is + =
Here is the tree diagram for rolling the 4 pyramid dice.
1st die 2nd die 3rd die 4th die
W
B
Here is the tree diagram for rolling the 4 pyramid dice.
1st die 2nd die 3rd die 4th die
W
W
B
B
W
B
Here is the tree diagram for rolling the 4 pyramid dice.
1st die 2nd die 3rd die 4th die
W
W
B
W
B
W
B
B
W
B
W
B
W
B
Here is the tree diagram for rolling the 4 pyramid dice.
1st die 2nd die 3rd die 4th die
W
W
B
W
B
W
B
WB
WB
WB
WB
B
W
B
W
B
W
B
WB
WB
WB
WB
Here is the tree diagram for rolling the 4 pyramid dice.
1st die 2nd die 3rd die 4th die
Since each arrow has a and we multiply along a branch to find
the chance of that branch occurring, each branch line has a
probability of × × × =
Follow 2 branch lines to see how this happens.
1st die 2nd die 3rd die 4th die
𝟏𝟐
W
W
B
1st die 2nd die 3rd die 4th die
𝟏𝟐
𝟏𝟐
𝟏𝟐
×
×
W
W
B
W
B
W
B
1st die 2nd die 3rd die 4th die
𝟏𝟐
𝟏𝟐
𝟏𝟐
𝟏𝟐
𝟏𝟐
×
××
×
W
W
B
W
B
W
B
WB
WB
WB
WB
1st die 2nd die 3rd die 4th die
𝟏𝟐
𝟏𝟐
𝟏𝟐
𝟏𝟐
𝟏𝟐
𝟏𝟐
𝟏𝟐
×
××
×
××
W
W
B
W
B
W
B
WB
WB
WB
WB
1st die 2nd die 3rd die 4th die
𝟏𝟐
𝟏𝟐
𝟏𝟐
𝟏𝟐
𝟏𝟐
𝟏𝟐
𝟏𝟐
×
××
× ¿ 𝟏𝟏𝟔
×× ¿ 𝟏
𝟏𝟔
W
W
B
W
B
W
B
WB
WB
WB
WB
1st die 2nd die 3rd die 4th die
𝟏𝟐
𝟏𝟐
𝟏𝟐
𝟏𝟐
𝟏𝟐
𝟏𝟐
𝟏𝟐
×
××
× ¿ 𝟏𝟏𝟔
×× ¿ 𝟏
𝟏𝟔
The probability of each branch occurring can be
written like this:P(W, W, W, W) = P(W, B, B, B) =
This can be done for each branch of the tree.
Only one branch of the tree will give a roll of 4. So the chance of rolling a 4 will always be .
W
W
B
W
B
W
B
WB
WB
WBWB
B
W
B
W
B
W
B
WB
WB
WBWB
1st die 2nd die 3rd die 4th die
But what about the chance of rolling a 2? There are 6 branches that have 2 whites (a score of 2).
W
W
B
W
B
W
B
WB
WB
WBWB
B
W
B
W
B
W
B
WB
WB
WBWB
1st die 2nd die 3rd die 4th die
116
116116116116116
But what about the chance of rolling a 2? There are 6 branches that have 2 whites (a score of 2).To find the probability of rolling a 2, add the chances of each of the 6 branches.
W
W
B
W
B
W
B
WB
WB
WBWB
B
W
B
W
B
W
B
WB
WB
WBWB
1st die 2nd die 3rd die 4th die
116
116116116116116
The chance of rolling a score of 2 is:
+ + + + + =
W
W
B
W
B
W
B
WB
WB
WBWB
B
W
B
W
B
W
B
WB
WB
WBWB
1st die 2nd die 3rd die 4th die
116
116116116116116
By doing this for each of the possibleoutcomes (0, 1, 2, 3 or 4) the probabilities for each option turn out to be: P(0) =
P(1) =
P(2) =
P(3) =
P(4) =
So our original question was:
Is there an EQUAL chance of these scores happening?
The answer: NO!
Now see if knowing these probabilities will help you in your
game.
Go to the link below to the British Museum website and play the
Royal Game of Ur online. (Requires Shockwave)
© Trustees of the British Museum
http://www.mesopotamia.co.uk/tombs/challenge/cha_set.html
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