You roll a fair six sided die (all 6 results are equally likely) 5 times independently. Let X be the number that roll results in 2 or 3. Find numerical value of a): pX (2.5) b): pX (1)
a) To find the probability of rolling a 2 or 3, let's first find the probability of rolling a 2 or 3 on a single roll: pX(2) + pX(3) = 1/6 + 1/6 = 1/3
Since the rolls are independent, the probability of rolling a 2 or 3 on 5 rolls is the same as the probability of rolling a 2 or 3 on a single roll raised to the power of 5:
pX(2.5) = (1/3)^5 = 1/243
b) To find the probability of rolling a 1 (not 2 or 3), we need to calculate pX(1):
pX(1) = 1 - pX(2) - pX(3) = 1 - 1/3 = 2/3
Since the rolls are independent, the probability of rolling a 1 on 5 rolls is the same as the probability of rolling a 1 on a single roll raised to the power of 5:
pX(1) = (2/3)^5 = 32/243