Successfully reported this slideshow.
We use your LinkedIn profile and activity data to personalize ads and to show you more relevant ads. You can change your ad preferences anytime.
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.
The Royal Game of Ur
had 5 playing pieces
And 4 dice per player.
Β© 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...
So the probability of rolling 1 (if a
white corner counts as a 1) is
𝑃 π‘€β„Žπ‘–π‘‘π‘’ =
2
4
=
𝟏
𝟐
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 h...
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 ...
In the tree diagram there will be 4
events matching the results of the
4 dice thrown.
W
B
First dice
𝑃 π‘€β„Žπ‘–π‘‘π‘’ =
2
4
=
𝟏
𝟐
𝑃...
W
W
B
B
W
B
1st dice 2nd dice
1
2
1
2
1
2
1
2
1
2
1
2
When the 2nd pyramid
is rolled, once again there
is a 1 chance in 2 ...
W
W
B
B
W
B
1st dice 2nd dice
1
2
1
2
1
2
1
2
1
2
1
2
The same can be done for
each branch of the tree.
Γ—
Probability of r...
W
W
B
B
W
B
1st dice 2nd dice
1
2
1
2
1
2
1
2
1
2
1
2
The same can be done for
each branch of the tree.
Γ—
Probability of r...
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
W
B
W
B
W
B
W
B
B
W
B
W
B
W
B
W
B
W
B
W
B
W
B
Here is the tree diagram for rolling
the 4 pyramid dice.
1st d...
Since each arrow has a
probability of
1
2
and we multiply
along a branch to find the chance
of that branch occurring, each...
1st die 2nd die 3rd die 4th die
𝟏
𝟐
W
W
B
1st die 2nd die 3rd die 4th die
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
Γ—
Γ—
W
W
B
W
B
W
B
1st die 2nd die 3rd die 4th die
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
Γ—
Γ—
Γ—
Γ—
W
W
B
W
B
W
B
W
B
W
B
W
B
W
B
1st die 2nd die 3rd die 4th die
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
Γ—
Γ—
Γ—
Γ—
Γ—
Γ—
W
W
B
W
B
W
B
W
B
W
B
W
B
W
B
1st die 2nd die 3rd die 4th die
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
Γ—
Γ—
Γ—
Γ— =
𝟏
πŸπŸ”
Γ—
Γ— =
𝟏
πŸπŸ”
W
W
B
W
B
W
B
W
B
W
B
W
B
W
B
1st die 2nd die 3rd die 4th die
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
Γ—
Γ—
Γ—
Γ— =
𝟏
πŸπŸ”
Γ—
Γ— =
𝟏
πŸπŸ”
The pr...
Only one branch of the tree will give a roll of 4.
So the chance of rolling a 4 will always be
1
16
.
W
W
B
W
B
W
B
W
B
W
...
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
W
B
W
B
W
...
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 ...
The chance of rolling a score of 2 is:
1
16
+
1
16
+
1
16
+
1
16
+
1
16
+
1
16
=
6
16
W
W
B
W
B
W
B
W
B
W
B
W
B
W
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) =...
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 ...
Upcoming SlideShare
Loading in …5
×

Probability in and Ancient Game

1,315 views

Published on

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

  • Be the first to comment

  • Be the first to like this

Probability in and Ancient Game

  1. 1. A look at chance and probability in the Royal Game of Ur
  2. 2. In 1922 C. Leonard Woolley discovered a 4000 year old game in a tomb in Ancient Ur.
  3. 3. The Royal Game of Ur had 5 playing pieces And 4 dice per player. Β© Trustees of the British Museum
  4. 4. 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. 𝑃 π‘€β„Žπ‘–π‘‘π‘’ = 2 4 = 𝟏 𝟐
  5. 5. So the probability of rolling 1 (if a white corner counts as a 1) is 𝑃 π‘€β„Žπ‘–π‘‘π‘’ = 2 4 = 𝟏 𝟐
  6. 6. 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?
  7. 7. 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.
  8. 8. In the tree diagram there will be 4 events matching the results of the 4 dice thrown. W B First dice 𝑃 π‘€β„Žπ‘–π‘‘π‘’ = 2 4 = 𝟏 𝟐 𝑃 π‘π‘™π‘Žπ‘π‘˜ = 2 4 = 𝟏 𝟐 1 2 1 2 So the chance of rolling a 1 (a white tip) is 1 chance in 2.
  9. 9. W W B B W B 1st dice 2nd dice 1 2 1 2 1 2 1 2 1 2 1 2 When the 2nd pyramid is rolled, once again there is a 1 chance in 2 of getting Another white tip. The chance of getting two white tips (rolling a 2) is found by multiplying along the branch of the tree diagram. This means that the chance of rolling a 2 is This is 1 chance in 4. Γ— 1 2 Γ— 1 2 = 𝟏 πŸ’
  10. 10. W W B B W B 1st dice 2nd dice 1 2 1 2 1 2 1 2 1 2 1 2 The same can be done for each branch of the tree. Γ— Probability of rolling a 2 is 1 4 Probability of rolling a 1 is 1 4 Probability of rolling a 1 is 1 4 Probability of rolling a 0 is 1 4
  11. 11. W W B B W B 1st dice 2nd dice 1 2 1 2 1 2 1 2 1 2 1 2 The same can be done for each branch of the tree. Γ— Probability of rolling a 2 is 1 4 Probability of rolling a 1 is 1 4 Probability of rolling a 1 is 1 4 Probability of rolling a 0 is 1 4 Since there are two possibilities of rolling a 1, we add the fractions. This gives a 2 4 or 1 2 chance of rolling a 1. The combined probability of rolling a 1 is 1 4 + 1 4 = 1 2
  12. 12. Here is the tree diagram for rolling the 4 pyramid dice. 1st die 2nd die 3rd die 4th die
  13. 13. W B Here is the tree diagram for rolling the 4 pyramid dice. 1st die 2nd die 3rd die 4th die
  14. 14. 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
  15. 15. 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
  16. 16. W W B W B W B W B W B W B W B B W B W B W B W 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
  17. 17. Since each arrow has a probability of 1 2 and we multiply along a branch to find the chance of that branch occurring, each branch line has a probability of 1 2 Γ— 1 2 Γ— 1 2 Γ— 1 2 = 1 16 . Follow 2 branch lines to see how this happens.
  18. 18. 1st die 2nd die 3rd die 4th die 𝟏 𝟐
  19. 19. W W B 1st die 2nd die 3rd die 4th die 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 Γ— Γ—
  20. 20. W W B W B W B 1st die 2nd die 3rd die 4th die 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 Γ— Γ— Γ— Γ—
  21. 21. W W B W B W B W B W B W B W B 1st die 2nd die 3rd die 4th die 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 Γ— Γ— Γ— Γ— Γ— Γ—
  22. 22. W W B W B W B W B W B W B W B 1st die 2nd die 3rd die 4th die 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 Γ— Γ— Γ— Γ— = 𝟏 πŸπŸ” Γ— Γ— = 𝟏 πŸπŸ”
  23. 23. W W B W B W B W B W B W B W B 1st die 2nd die 3rd die 4th die 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 𝟏 𝟐 Γ— Γ— Γ— Γ— = 𝟏 πŸπŸ” Γ— Γ— = 𝟏 πŸπŸ” The probability of each branch occurring can be written like this: P(W, W, W, W) = 1 16 P(W, B, B, B) = 1 16 This can be done for each branch of the tree.
  24. 24. Only one branch of the tree will give a roll of 4. So the chance of rolling a 4 will always be 1 16 . W W B W B W B W B W B W B W B B W B W B W B W B W B W B W B 1st die 2nd die 3rd die 4th die
  25. 25. 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 W B W B W B W B B W B W B W B W B W B W B W B 1st die 2nd die 3rd die 4th die 1 16 1 16 1 16 1 16 1 16 1 16
  26. 26. 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 W B W B W B W B B W B W B W B W B W B W B W B 1st die 2nd die 3rd die 4th die 1 16 1 16 1 16 1 16 1 16 1 16
  27. 27. The chance of rolling a score of 2 is: 1 16 + 1 16 + 1 16 + 1 16 + 1 16 + 1 16 = 6 16 W W B W B W B W B W B W B W B B W B W B W B W B W B W B W B 1st die 2nd die 3rd die 4th die 1 16 1 16 1 16 1 16 1 16 1 16
  28. 28. 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) = 1 16 P(1) = 4 16 P(2) = 6 16 P(3) = 4 16 P(4) = 1 16
  29. 29. So our original question was: Is there an EQUAL chance of these scores happening? The answer: NO!
  30. 30. 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) Β© Trustees of the British Museum http://www.mesopotamia.co.uk/tombs/challenge/cha_set.html

Γ—