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Let's break down the problem step by step:

We are given that:
- The tennis balls in the bag come in two colors: red and green.
- The ratio of red balls to green balls is 1:3.
- Two balls are picked at random, one after the other, without replacement.
- The probability of picking a red ball first and a green ball second is \( \frac{1}{5} \).

We are asked to find the total number of balls in the bag.

### Step 1: Let the number of red balls be \( x \). Since the ratio of red to green is 1:3, the number of green balls will be \( 3x \). Therefore, the total number of balls in the bag is \( x + 3x = 4x \).

### Step 2: The probability of picking a red ball first is:
\[
P(\text{Red first}) = \frac{x}{4x} = \frac{1}{4}
\]

### Step 3: After picking one red ball, the number of balls left in the bag is \( 4x - 1 \), and the number of green balls left is still \( 3x \). The probability of picking a green ball second is:
\[
P(\text{Green second}) = \frac{3x}{4x - 1}
\]

### Step 4: The total probability of picking a red ball first and a green ball second is the product of the two probabilities:
\[
P(\text{Red then Green}) = \frac{1}{4} \times \frac{3x}{4x - 1}
\]

We are told that this probability is \( \frac{1}{5} \). So, we can set up the equation:
\[
\frac{1}{4} \times \frac{3x}{4x - 1} = \frac{1}{5}
\]

### Step 5: Solve the equation
\[
\frac{3x}{4(4x - 1)} = \frac{1}{5}
\]

Cross-multiply:
\[
5 \times 3x = 4 \times (4x - 1)
\]

Simplify:
\[
15x = 16x - 4
\]

Rearrange:
\[
x = 4
\]

### Step 6: Conclusion
Since \( x = 4 \), the total number of balls in the bag is \( 4x = 16 \).

### Final Answer:
There are 16 balls in the bag.

Let me know if you'd like further details or clarifications.

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**Related Questions:**
1. What would happen to the probability if the ratio of red to green balls was 2:3 instead?
2. How does the probability change if the balls were picked with replacement?
3. What is the probability of picking two red balls in succession without replacement?
4. How many red balls are there in the bag if the total is 16?
5. Can the problem be solved if the ratio was unknown, but the probability was given?

**Tip:** When working with probabilities involving "without replacement," always remember that the total number of items and the specific number of each type decrease after each selection.

A bag contains only two colours of tennis balls, red and green, in the ratio 1:3. Two balls are picked at random, one after the other, without replacement. Calculate the number of balls in the bag given that the probability of picking first a red ball and second a green ball is equal to 1/5.

Learn how to calculate the probability of selecting a red ball first and a green ball second from a bag containing red and green balls in the ratio 1:3. This detailed solution covers key probability concepts and the multiplication rule for independent events. Suitable for students in Grades 10-12.

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