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This presentation looks at the dice used in the 4000 year old Royal Game of Ur discovered in the ...

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

- A look at chance and probability

in the

Royal Game of Ur - In 1922 C. Leonard Woolley

discovered a 4000 year old game

in a tomb in Ancient Ur. - © Trustees of the British Museum

The Royal Game of Ur

had 5 playing pieces

And 4 dice per player. - 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.

B

This would count as 0.

W - In the tree diagram there will be 4

events matching the results of the

4 dice thrown.

First dice

W

=

So the chance of rolling a 1 (a white tip)

is 1 chance in 2.

=

B - 1st dice

2nd dice

When the 2nd pyramid

is rolled, once again there

is a 1 chance in 2 of getting

W

Another white tip.

W

The chance of getting two

white tips (rolling a 2) is

B

found by multiplying along

the branch of the tree

diagram.

W

This means that the chance

of rolling a 2 is

B

B

This is 1 chance in 4. - 1st dice

2nd dice

The same can be done for

each branch of the tree.

W Probability of rolling a 2 is

W

B Probability of rol ing a 1 is

W Probability of rolling a 1 is

B

B

Probability of rolling a 0 is - 1st dice

2nd dice

The same can be done for

each branch of the tree.

W Probability of rolling a 2 is

W

Since there

are two

B Probability of rol ing a 1 is

possibilities

of rolling a 1,

T he combined probability of

we add the

rolling a 1 is + =

fractions.

This gives a

W Probability of rolling a 1 is

or chance

of rolling a 1.

B

B

Probability of rolling a 0 is - Here is the tree diagram for rolling

the 4 pyramid dice.

1st die

2nd die

3rd die

4th die - 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

W

B

B - Here is the tree diagram for rolling

the 4 pyramid dice.

1st die

2nd die

3rd die

4th die

W

W

B

W

W

B

B

W

W

B

B

W

B

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

W

W

W

B

B

B

W

B

W

W

B

W

B

W

B

B

W

W

B

B

B

W

B - 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 - 1st die

2nd die

3rd die

4th die

W

W

B - 1st die

2nd die

3rd die

4th die

W

W

B

W

W

B

B - 1st die

2nd die

3rd die

4th die

W

W

B

W

B

W

B

W

W

W

B

B

B

W

B - 1st die

2nd die

3rd die

4th die

W

W

B

W

B

W

B

W

W

W

B

B

B

W

B - 1st die

2nd die

3rd die

4th die

W

W

B

W

B

W

B

W

W

W

B

B

B

W

B

T he 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. - O

nly one branch of the tree will give a roll of 4.

So the chance of rolling a 4 will always be .

1st die

2nd die 3rd die 4th die

W

W

B

W

B

W

B

W

W

W

B

B

B

W

B

W

W

B

W

B

W

B

B

W

W

B

B

B

W

B - But what about the chance of rolling a 2? There

are 6 branches that have 2 whites (a score of 2).

1st die

2nd die 3rd die 4th die

W

W

B

W

B

W

B

W

W

W

B

B

B

W

B

W

W

B

W

B

W

B

B

W

W

B

B

B

W

B - 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.

1st die

2nd die 3rd die 4th die

W

W

B

W

B

W

B

W

W

W

B

B

B

W

B

W

W

B

W

B

W

B

B

W

W

B

B

B

W

B - T he chance of rolling a score of 2 is:

+ + + + + =

1st die

2nd die 3rd die 4th die

W

W

B

W

B

W

B

W

W

W

B

B

B

W

B

W

W

B

W

B

W

B

B

W

W

B

B

B

W

B - By doing this for each of the possible

outcomes (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

wil help you in your game.

© TrustG

ees o

of the t

B o

ritish t

M h

useue

m

link below to the British

Museum website and play the

Royal Game of Ur online.

(Requires Shockwave)

http://www.mesopotamia.co.uk/tombs/chal enge/cha_set.html