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?

Question

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?

UpStudy AI · Accepted Answer

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
$$\boxed{\dfrac{55}{216}}$$
The probability is $$ \frac{55}{216} $$.

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?

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