1. Three Dice - An Exploration of Conditional Probability

2. 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}.

3. 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

4. Then toss the selected die. If we know the die that we are using, we can conditionally state the probabilities for each face value. FaceGiven d4 Given d6 Given d8 10.25000.16670.1250 20.25000.16670.1250 30.25000.16670.1250 40.25000.16670.1250 50.00000.16670.1250 60.00000.16670.1250 70.00000.00000.1250 80.00000.00000.1250 Total1.00001.00001.0000

5. 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.

6. 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})

7. 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