Step 1: Understanding the problem:
We are given that a bag contains 4 red, 5 white, and some yellow balls. The probability of drawing a red ball at random is given as \( \frac{1}{5} \). We need to find the probability of drawing a yellow ball.
Step 2: Finding the total number of balls in the bag:
Let the number of yellow balls be \( y \). The total number of balls in the bag is the sum of red, white, and yellow balls, so the total number of balls is:
\[
\text{Total balls} = 4 \, (\text{red}) + 5 \, (\text{white}) + y \, (\text{yellow}) = 9 + y
\]
The probability of drawing a red ball is given by the ratio of red balls to the total number of balls. This probability is \( \frac{1}{5} \), so we have the equation:
\[
\frac{4}{9 + y} = \frac{1}{5}
\]
Step 3: Solve for \( y \):
To solve for \( y \), cross-multiply the equation:
\[
4 \times 5 = 1 \times (9 + y)
\]
\[
20 = 9 + y
\]
Now, subtract 9 from both sides:
\[
y = 20 - 9 = 11
\]
Thus, the number of yellow balls is \( y = 11 \).
Step 4: Finding the probability of drawing a yellow ball:
Now that we know there are 11 yellow balls, the total number of balls in the bag is:
\[
\text{Total balls} = 9 + 11 = 20
\]
The probability of drawing a yellow ball is the ratio of yellow balls to the total number of balls:
\[
\text{Probability of yellow ball} = \frac{11}{20}
\]
Step 5: Conclusion:
The probability of drawing a yellow ball at random is \( \frac{11}{20} \).