A fair six-sided dice is rolled three times. What is the probability that the outcome of the first roll is lower than the outcomes of the second and third roll?

To determine the probability that the outcome of the first roll is lower than the outcomes of the second and third rolls when rolling a fair six-sided die three times, we can follow these steps: ### Total Possible Outcomes Each die roll has 6 possible outcomes. Since the die is rolled three times, the total number of possible outcomes is: \[ 6 \times 6 \times 6 = 216 \] ### Favorable Outcomes We need to count the number of outcomes where the first roll \( a \) is less than both the second roll \( b \) and the third roll \( c \), i.e., \( a < b \) and \( a < c \). Consider each possible value of \( a \) (from 1 to 6): 1. **If \( a = 1 \):** - \( b \) and \( c \) can each be any of {2, 3, 4, 5, 6}. - Number of possibilities: \( 5 \times 5 = 25 \). 2. **If \( a = 2 \):** - \( b \) and \( c \) can each be any of {3, 4, 5, 6}. - Number of possibilities: \( 4 \times 4 = 16 \). 3. **If \( a = 3 \):** - \( b \) and \( c \) can each be any of {4, 5, 6}. - Number of possibilities: \( 3 \times 3 = 9 \). 4. **If \( a = 4 \):** - \( b \) and \( c \) can each be any of {5, 6}. - Number of possibilities: \( 2 \times 2 = 4 \). 5. **If \( a = 5 \):** - \( b \) and \( c \) can each be 6. - Number of possibilities: \( 1 \times 1 = 1 \). 6. **If \( a = 6 \):** - There are no possible values for \( b \) and \( c \) that are greater than 6. - Number of possibilities: 0. Adding these up: \[ 25 + 16 + 9 + 4 + 1 = 55 \] ### Probability Calculation The probability is the ratio of favorable outcomes to total possible outcomes: \[ \text{Probability} = \frac{55}{216} \] ### Final Answer \[\boxed{\dfrac{55}{216}}\]