1. Key Concepts and Formulas

• Quadratic Equation and Discriminant: For a quadratic equation of the form $ax^2 + bx + c = 0$ (where $a \neq 0$), the nature of its roots is determined by the discriminant, $D$. The formula for the discriminant is: $D = b^2 - 4ac$
• Condition for Equal Roots: The quadratic equation has equal roots if and only if its discriminant is zero. $D = 0 \implies b^2 - 4ac = 0 \implies b^2 = 4ac$
• Classical Probability: The probability of an event $E$ occurring is given by the ratio of the number of favorable outcomes $n(E)$ to the total number of possible outcomes in the sample space $n(S)$, assuming all outcomes are equally likely: $P(E) = \frac{n(E)}{n(S)}$

2. Step-by-Step Solution

Step 1: Determine the Sample Space (Total Number of Outcomes)

We are given that the coefficients $a$, $b$, and $c$ of the quadratic equation are obtained by throwing a standard six-sided dice three times. This means each coefficient can independently take any integer value from 1 to 6.

• The possible values for $a$ are $\{1, 2, 3, 4, 5, 6\}$.
• The possible values for $b$ are $\{1, 2, 3, 4, 5, 6\}$.
• The possible values for $c$ are $\{1, 2, 3, 4, 5, 6\}$.

Since each dice throw is an independent event, the total number of unique combinations of $(a, b, c)$ is found by multiplying the number of choices for each coefficient. This forms our sample space, $n(S)$. $n(S) = 6 \times 6 \times 6 = 216$ Each of these 216 combinations is equally likely.

Step 2: Identify Favorable Outcomes (Cases where $b^2 = 4ac$)

Our goal is to find all combinations of $(a, b, c)$ from our sample space (where $a, b, c \in \{1, 2, 3, 4, 5, 6\}$) that satisfy the condition for equal roots: $b^2 = 4ac$.

Crucial Observation to Simplify the Search: Let's analyze the equation $b^2 = 4ac$. The right-hand side, $4ac$, is always a multiple of 4 (since it has a factor of 4). This means that $b^2$ must also be a multiple of 4. This property implies that $b$ must be an even number. If $b$ is odd, $b^2$ is odd and cannot be a multiple of 4. If $b$ is even, say $b=2k$, then $b^2 = (2k)^2 = 4k^2$, which is always a multiple of 4. Therefore, for $b^2 = 4ac$ to hold, $b$ must be an even number. This significantly reduces the number of cases we need to check, as we only need to consider $b \in \{2, 4, 6\}$.

Let's systematically check each possible even value for $b$:

• Case 1: $b = 2$ Substitute $b=2$ into the condition $b^2 = 4ac$: $2^2 = 4ac \implies 4 = 4ac \implies ac = 1$ Now we need to find pairs of $(a, c)$ such that their product is 1, and $a, c \in \{1, 2, 3, 4, 5, 6\}$. The only possible pair is $a=1$ and $c=1$. So, one favorable outcome is $(a, b, c) = (1, 2, 1)$.

• Case 2: $b = 4$ Substitute $b=4$ into the condition $b^2 = 4ac$: $4^2 = 4ac \implies 16 = 4ac \implies ac = 4$ Now we need to find pairs of $(a, c)$ such that their product is 4, and $a, c \in \{1, 2, 3, 4, 5, 6\}$. The pair $(a,c) = (2,2)$ satisfies this condition. So, one favorable outcome is $(a, b, c) = (2, 4, 2)$.

• Case 3: $b = 6$ Substitute $b=6$ into the condition $b^2 = 4ac$: $6^2 = 4ac \implies 36 = 4ac \implies ac = 9$ Now we need to find pairs of $(a, c)$ such that their product is 9, and $a, c \in \{1, 2, 3, 4, 5, 6\}$. The only possible pair is $a=3$ and $c=3$. (Pairs like $(1,9)$ are invalid as $c=9$ is not a dice outcome). So, one favorable outcome is $(a, b, c) = (3, 6, 3)$.

Let $n(E)$ be the total number of favorable outcomes. Summing the outcomes from all valid cases: $n(E) = 1 \text{ (from } b=2) + 1 \text{ (from } b=4) + 1 \text{ (from } b=6) = 3$

Step 3: Calculate the Probability

The probability of an event $E$ (having equal roots) is given by the ratio of the number of favorable outcomes to the total number of possible outcomes: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{n(E)}{n(S)}$ Substituting the values we found: $P(E) = \frac{3}{216}$ This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3: $P(E) = \frac{3 \div 3}{216 \div 3} = \frac{1}{72}$

3. Common Mistakes & Tips

• Understand the Discriminant: Make sure you correctly recall the condition for equal roots ($D=0$). This is the cornerstone of the problem.
• Range of Coefficients: Always remember that $a, b, c$ are outcomes of a standard dice roll, meaning they must be integers from 1 to 6, inclusive. This is crucial when filtering possible pairs for $ac$.
• Systematic Approach & Mathematical Properties: Utilize the observation that $b$ must be an even number for $b^2 = 4ac$ to hold. This significantly reduces the number of cases to check. Systematically check each possible value of $b$.

4. Summary

To determine the probability that a quadratic equation, whose coefficients are determined by dice throws, has equal roots, we followed a systematic approach:

1. We identified the condition for equal roots as $b^2 = 4ac$.
2. We calculated the total sample space: since each of the three coefficients can be any integer from 1 to 6, the total number of possible $(a,b,c)$ combinations is $6^3 = 216$.
3. We found favorable outcomes by observing that $b^2 = 4ac$ implies $b$ must be an even number. We then systematically checked $b=2, 4, 6$. For each valid $b$, we found pairs of $(a,c)$ that satisfy $ac = b^2/4$ and are within the range $\{1, \dots, 6\}$. This yielded 3 favorable outcomes: $(1,2,1)$, $(2,4,2)$, and $(3,6,3)$.
4. Finally, we calculated the probability as the ratio of favorable outcomes to the total outcomes, which is $\frac{3}{216} = \frac{1}{72}$.

5. Final Answer

The final answer is $\boxed{{1 \over {72}}}$, which corresponds to option (A).