If the probability of winning a race of an athlete is `1/6` less than the twice the probability of losing the race. Find the probability of winning the race.

Options

A. `7//18`
B. `11//18`
C. `5//18`
D. `3//18`

Answer

To solve the problem step by step, we will define the variables and set up the equations based on the information given in the question.

### Step 1: Define Variables
Let the probability of winning the race be denoted as \( x \).

### Step 2: Express the Probability of Losing
Since there are only two outcomes (winning or losing), the probability of losing the race can be expressed as:
\[ \text{Probability of losing} = 1 - x \]

### Step 3: Set Up the Equation
According to the problem, the probability of winning is \( \frac{1}{6} \) less than twice the probability of losing. This can be expressed mathematically as:
\[ x = 2(1 - x) - \frac{1}{6} \]

### Step 4: Simplify the Equation
Now, let's simplify the equation:
\[ x = 2(1 - x) - \frac{1}{6} \]
Expanding the right side:
\[ x = 2 - 2x - \frac{1}{6} \]

### Step 5: Combine Like Terms
To eliminate the fraction, we can multiply the entire equation by 6 to make calculations easier:
\[ 6x = 6(2 - 2x - \frac{1}{6}) \]
This simplifies to:
\[ 6x = 12 - 12x - 1 \]
Combining the constants gives:
\[ 6x = 11 - 12x \]

### Step 6: Solve for \( x \)
Now, we will add \( 12x \) to both sides:
\[ 6x + 12x = 11 \]
This simplifies to:
\[ 18x = 11 \]
Now, divide both sides by 18:
\[ x = \frac{11}{18} \]

### Step 7: Conclusion
The probability of winning the race is:
\[ \boxed{\frac{11}{18}} \]