To solve the problem, we need to determine the probability that the roots of the quadratic equation \( ax^2 + bx + c = 0 \) are imaginary, given that the coefficients \( a \), \( b \), and \( c \) are determined by tossing three fair coins where 'head' represents 1 and 'tail' represents 2.
### Step-by-Step Solution:
1. **Understanding the Discriminant**:
The roots of the quadratic equation are imaginary if the discriminant \( D \) is less than 0. The discriminant is given by:
\[
D = b^2 - 4ac
\]
We need to find when \( D < 0 \).
2. **Sample Space**:
When tossing three coins, each coin can either be a head (1) or a tail (2). Therefore, the total number of outcomes when tossing three coins is \( 2^3 = 8 \). The possible outcomes for \( (a, b, c) \) based on the coin tosses are:
- (1, 1, 1)
- (1, 1, 2)
- (1, 2, 1)
- (1, 2, 2)
- (2, 1, 1)
- (2, 1, 2)
- (2, 2, 1)
- (2, 2, 2)
3. **Calculating the Discriminant for Each Case**:
We will evaluate \( D = b^2 - 4ac \) for each combination:
- For \( (1, 1, 1) \): \( D = 1^2 - 4(1)(1) = 1 - 4 = -3 \) (Imaginary)
- For \( (1, 1, 2) \): \( D = 1^2 - 4(1)(2) = 1 - 8 = -7 \) (Imaginary)
- For \( (1, 2, 1) \): \( D = 2^2 - 4(1)(1) = 4 - 4 = 0 \) (Not Imaginary)
- For \( (1, 2, 2) \): \( D = 2^2 - 4(1)(2) = 4 - 8 = -4 \) (Imaginary)
- For \( (2, 1, 1) \): \( D = 1^2 - 4(2)(1) = 1 - 8 = -7 \) (Imaginary)
- For \( (2, 1, 2) \): \( D = 1^2 - 4(2)(2) = 1 - 16 = -15 \) (Imaginary)
- For \( (2, 2, 1) \): \( D = 2^2 - 4(2)(1) = 4 - 8 = -4 \) (Imaginary)
- For \( (2, 2, 2) \): \( D = 2^2 - 4(2)(2) = 4 - 16 = -12 \) (Imaginary)
4. **Counting Favorable Outcomes**:
From the evaluations:
- Imaginary roots occur for the following combinations:
- (1, 1, 1)
- (1, 1, 2)
- (1, 2, 2)
- (2, 1, 1)
- (2, 1, 2)
- (2, 2, 1)
- (2, 2, 2)
- Total favorable outcomes = 7.
5. **Calculating the Probability**:
The probability \( P \) that the roots are imaginary is given by the ratio of favorable outcomes to the total outcomes:
\[
P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{7}{8}
\]
### Final Answer:
The probability that the roots of the equation are imaginary is \( \frac{7}{8} \).
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