Three Dice - An Exploration of Conditional Probability

Consider three dice, all fair:

 

Four-sided die (d4) with faces {1,2,3,4}

Six-sided die (d6) with faces {1,2,3,4,5,6}

Eight-sided die (d8) with faces {1,2,3,4,5,6,7,8}.

Consider a two-stage experiment:

First, select a die at random – 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

Then toss the selected die. 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