Answer:
a) The probability of a tie is the probability of getting the same outcome in both dice.
First, we roll die A and we get a given outcome. (here the probability is 1 because there is no restriction)
Now we need to roll die B and get the same outcome, so here we have only one possible outcome out of 6, then in this case the probability is 1/6
The joint probability (the product of the individual probabilities) is: P = 1*(1/6) = 1/6
b) We want to find the probability that die A wins.
Here we need to analyze each possible outcome for die A.
Let's define:
pₙ = probability of winning given that the outcome is n.
If the outcome is a 1, die A can only lose or tie, then here:
p₁ = 0
If the outcome is a 2, then die A only wins if die B rolls a 1, (1 outcome out of 6)
Then the probability of winning is:
p₂ = 1/6
If the outcome is a 3, then die A wins if die B rolls a 1 or a 2, here we have two possible outcomes out of 6.
p₃ = 2/6
We already can see the pattern, if the outcome is a 4, we get:
p₄ = 3/6
if the outcome is a 5:
p₅ = 4/6
if the outcome is a 6:
p₆ = 5/6
The total probability is the sum of all the joint probabilities:
For each roll in dice A, the probability is 1/6 (the probability of getting a given outcome)
Then the probability that die A wins is:
P = (1/6)*p₁ + (1/6)*p₂ + ... = (1/6)*(p₁ + p₂ + p₃ + p₄ + p₅ + p₆)
P = (1/6)*(0 + 1/6 + 2/6 + 3/6 + 4/6 + 5/6)
P = (1/6)*(15/6) = 0.4167