I just wrote a question and I'm not sure if my solution is correct. Could someone please help me check it?
Question: Mary starts at 0 on the number line. She first tosses a coin, and then rolls a fair six-sided dice. If the coin lands on heads, then she moves to the positive side; if the coin lands on tails, then she moves to the negative side. The number of steps she takes depends on the number that appears on the dice she rolled. For example, if the coin lands on heads and 4 is rolled, then Mary would go 4 steps to the positive side. What is the probability that Mary will end at 0 after three rounds of tossing and rolling?
My solution: In order for Mary to end at 0, one of the dice must be the sum of the other two dice and be the opposite sign.
P (sum = 2) = ⅙ x ⅙ x ⅙ x 3
P (sum = 3) = ⅙ x ⅙ x ⅙ x 6
P (sum = 4) = ⅙ x ⅙ x ⅙ x (3 + 6)
P (sum = 5) = ⅙ x ⅙ x ⅙ x (6 + 6)
P (sum = 6) = ⅙ x ⅙ x ⅙ x (3 + 6 + 6)
P total = ⅙ x ⅙ x ⅙ x 45 = 5/24.
There is 1/2 probability that the two dice are the same sign, and we multiply it by another 1/2 to ensure that the other dice is the opposite sign. Therefore my answer is 5/96.