It is not a good idea to go first because whichever die Charlie selects there is another die which has a higher probability of winning against him.
Alison can select her die using the following logic -
1. If Charlie selects the red die i.e. {1,1,6,6,8,8} then Alison should select the green die because –
a. The green die is {2,2,4,4,9,9}
i. The probability of Charlie rolling a 1 is 2 from 6 and the probability of Alison rolling a greater number is 6 from 6
ii. The probability of Charlie rolling a 6 is 2 from 6 and the probability of Alison rolling a greater number is 2 from 6
iii. The probability of Charlie rolling a 8 is 2 from 6 and the probability of Alison rolling a greater number is 2 from 6
Therefore the total probability of Alison winning is (6 + 2 + 2) = 10
b. The blue die is {3,3,5,5,7,7}
i. The probability of Charlie rolling a 1 is 2 from 6 and the probability of Alison rolling a greater number is 6 from 6
ii. The probability of Charlie rolling a 6 is 2 from 6 and the probability of Alison rolling a greater number is 2 from 6
iii. The probability of Charlie rolling a 8 is 2 from 6 and the probability of Alison rolling a greater number is 0 from 6
Therefore the total probability of Alison winning is (6 + 2 + 0) = 8
2. If Charlie selects the green die i.e. {2,2,4,4,9,9} then Alison should select the blue die because –
a. The red die is {1,1,6,6,8,8}
i. The probability of Charlie rolling a 2 is 2 from 6 and the probability of Alison rolling a greater number is 4 from 6
ii. The probability of Charlie rolling a 4 is 2 from 6 and the probability of Alison rolling a greater number is 4 from 6
iii. The probability of Charlie rolling a 9 is 2 from 6 and the probability of Alison rolling a greater number is 0 from 6
Therefore the total probability of Alison winning is (4 + 4 + 0) = 8
b. The blue die is {3,3,5,5,7,7}
i. The probability of Charlie rolling a 2 is 2 from 6 and the probability of Alison rolling a greater number is 6 from 6
ii. The probability of Charlie rolling a 4 is 2 from 6 and the probability of Alison rolling a greater number is 4 from 6
iii. The probability of Charlie rolling a 9 is 2 from 6 and the probability of Alison rolling a greater number is 0 from 6
Therefore the total probability of Alison winning is (6 + 4 + 0) = 10
3. If Charlie selects the blue die i.e. {3,3,5,5,7,7} then Alison should select the red die because –
a. The red die is {1,1,6,6,8,8}
i. The probability of Charlie rolling a 3 is 2 from 6 and the probability of Alison rolling a greater number is 4 from 6
ii. The probability of Charlie rolling a 5 is 2 from 6 and the probability of Alison rolling a greater number is 4 from 6
iii. The probability of Charlie rolling a 7 is 2 from 6 and the probability of Alison rolling a greater number is 2 from 6
Therefore the total probability of Alison winning is (4 + 4 + 2) = 10
b. The green die is {2,2,4,4,9,9}
i. The probability of Charlie rolling a 3 is 2 from 6 and the probability of Alison rolling a greater number is 4 from 6
ii. The probability of Charlie rolling a 5 is 2 from 6 and the probability of Alison rolling a greater number is 2 from 6
iii. The probability of Charlie rolling a 7 is 2 from 6 and the probability of Alison rolling a greater number is 2 from 6
Therefore the total probability of Alison winning is (4 + 2 + 2) = 8