Three Dice - An Exploration of Conditional Probability
Consider three dice, all fair:
d4 with
faces {1,2,3,4}
d6 with faces
{1,2,3,4,5,6}
d8 with faces
{1,2,3,4,5,6,7,8}.
Consider a two-step experiment: first,
select a die at random, then toss that die. Suppose that each of the dice has
an equal chance of being selected for each toss.
|
Pr{select d4}=1/3 |
|
Pr{select d6}=1/3 |
|
Pr{select d8}=1/3 |
If we know the die that we are using, we can conditionally state the
probabilities for each face value.
|
Face |
Given d4 |
Given d6 |
Given d8 |
||
|
1 |
0.2500 |
|
0.1667 |
|
0.1250 |
|
2 |
0.2500 |
|
0.1667 |
|
0.1250 |
|
3 |
0.2500 |
|
0.1667 |
|
0.1250 |
|
4 |
0.2500 |
|
0.1667 |
|
0.1250 |
|
5 |
0.0000 |
|
0.1667 |
|
0.1250 |
|
6 |
0.0000 |
|
0.1667 |
|
0.1250 |
|
7 |
0.0000 |
|
0.0000 |
|
0.1250 |
|
8 |
0.0000 |
|
0.0000 |
|
0.1250 |
|
Total |
1.0000 |
|
1.0000 |
|
1.0000 |
We have a specific way of writing conditional probabilities. For example:
|
Pr{1 shows | d4 selected} = 1/4 |
|
Pr{1 shows | d6 selected} = 1/6 |
|
Pr{1 shows | d8 selected} = 1/8 |
The “|” indicates the probability for the event on the left of the mark is being computed
under the assumption that the event on the right of the mark occurs with certainty.
The total probability for each face value, accounting for the
selection of the die and the die itself, depends on both the selection of the
die, and the results of the toss of the selected die.
The basic formula works like this:
|
Pr{face
shows} = |
Pr{face shows and d4 is selected}+ |
||
|
|
|
|
Pr{face shows and d6 is selected}+ |
|
|
|
|
Pr{face shows and d8 is selected} |
This is the same as:
|
Pr{face
shows}= |
Pr{d4 is selected}*Pr{face shows|d4 is selected}+ |
||
|
|
|
|
Pr{d6 is selected}*Pr{face shows|d6 is selected}+ |
|
|
|
|
Pr{d8 is selected}*Pr{face shows|d8 is selected} |
Computing probabilities for each face value:
|
Pr{1
shows} = (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) |
= |
0.1806 |
||||||
|
Pr{2
shows} = (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) |
= |
0.1806 |
||||||
|
Pr{3
shows} = (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) |
= |
0.1806 |
||||||
|
Pr{4
shows} = (1/3)*(1/4) + (1/3)*(1/6) + (1/3)*(1/8) |
= |
0.1806 |
||||||
|
Pr{5
shows} = (1/3)*(1/6) + (1/3)*(1/8) |
|
= |
0.0972 |
|||||
|
Pr{6
shows} = (1/3)*(1/6) + (1/3)*(1/8) |
|
= |
0.0972 |
|||||
|
Pr{7
shows} = (1/3)*(1/8) |
|
|
|
= |
0.0417 |
|||
|
Pr{8
shows} = (1/3)*(1/8) |
|
|
|
= |
0.0417 |
|||
|
Total |
|
|
|
|
|
|
= |
1.0000 |
Note that the d4 does not contribute any probability to faces
5,6,7,8. Note that the d6 does not contribute any probability to faces 7,8