Each question below contains a statement followed by Quantity I and Quantity II. Find both to find the relationship among them. Mark your answer accordingly.
Quantity I : A bag contains 50 balls which are green, orange and yellow . Number of orange balls in the bag, if probability of picking green ball is `3/5` and that of either a green or orange ball is `4/5`
Quantity II: A bag contains 40 balls green, orange and yellow. The probability of picking orange ball is `3/8` If the first ball was orange and without replacement, probability of picking a green ball is `4/13`. Number of yellow balls.

Options

A. Quantity I `gt` Quantity II
B. Quantity I `lt` Quantity II
C. Quantity I `ge` Quantity II
D. Quantity I `le` Quantity II

Explanation

To solve the problem, we will analyze both Quantity I and Quantity II step by step.

### Quantity I:
We know the following:
- Total number of balls = 50
- Probability of picking a green ball (P(G)) = 3/5
- Probability of picking either a green or orange ball (P(G or O)) = 4/5

1. **Calculate the number of green balls (G)**:
The probability of picking a green ball is given by:
\[ P(G) = \frac{\text{Number of green balls}}{\text{Total number of balls}} = \frac{G}{50} \]
Setting this equal to the given probability:
\[ \frac{G}{50} = \frac{3}{5} \]
Cross-multiplying gives:
\[ G = 50 \times \frac{3}{5} = 30 \]

2. **Calculate the number of orange balls (O)**:
The probability of picking either a green or orange ball is:
\[ P(G \text{ or } O) = \frac{G + O}{50} \]
Setting this equal to the given probability:
\[ \frac{G + O}{50} = \frac{4}{5} \]
Cross-multiplying gives:
\[ G + O = 50 \times \frac{4}{5} = 40 \]
Now substituting the value of G:
\[ 30 + O = 40 \implies O = 40 - 30 = 10 \]

So, the number of orange balls in Quantity I is **10**.

### Quantity II:
We know the following:
- Total number of balls = 40
- Probability of picking an orange ball (P(O)) = 3/8
- If the first ball was orange and without replacement, the probability of picking a green ball (P(G|O)) = 4/13

1. **Calculate the number of orange balls (O)**:
The probability of picking an orange ball is given by:
\[ P(O) = \frac{O}{40} \]
Setting this equal to the given probability:
\[ \frac{O}{40} = \frac{3}{8} \]
Cross-multiplying gives:
\[ O = 40 \times \frac{3}{8} = 15 \]

2. **Calculate the number of green balls (G)**:
After removing one orange ball, the total number of balls becomes 39. The probability of picking a green ball now is:
\[ P(G|O) = \frac{G}{39} \]
Setting this equal to the given probability:
\[ \frac{G}{39} = \frac{4}{13} \]
Cross-multiplying gives:
\[ G = 39 \times \frac{4}{13} = 12 \]

3. **Calculate the number of yellow balls (Y)**:
The total number of balls is 40, so:
\[ Y = 40 - (G + O) = 40 - (12 + 15) = 40 - 27 = 13 \]

So, the number of yellow balls in Quantity II is **13**.

### Summary of Results:
- Quantity I (Number of orange balls) = **10**
- Quantity II (Number of yellow balls) = **13**

### Conclusion:
Since 13 (Quantity II) is greater than 10 (Quantity I), we conclude that:
**Quantity II > Quantity I**.