Question

A box contains five balls of same size and shape. Three of them are green coloured balls and two of them are orange coloured balls. Balls are drawn from the box one at a time. If a green ball is drawn, it is not replaced. If an orange ball is drawn, it is replaced with another orange ball. First ball is drawn. What is the probability of getting an orange ball in the next draw?

• A. $\frac{1}{2}$
• B. $\frac{8}{25}$
• C. $\frac{19}{50}$
• D. $\frac{23}{50}$

Hint:

In probability problems involving replacement rules, treat each case separately and use total probability law to combine the results.

Answer 1

The Correct Option is D

Initially, the box contains 3 green balls and 2 orange balls, making a total of 5 balls.
We must compute the probability of drawing an orange ball on the second draw, considering the two possible outcomes of the first draw.
Case 1: First ball is green (G).
Probability of drawing green first = $\frac{3}{5}$.
If a green ball is drawn, it is \emph{not replaced}.
So the box now contains 2 green and 2 orange balls (4 total).
Probability of orange in next draw = $\frac{2}{4} = \frac{1}{2}$.
Contribution of this case:
\[ \frac{3}{5} \times \frac{1}{2} = \frac{3}{10} \]
Case 2: First ball is orange (O).
Probability of drawing orange first = $\frac{2}{5}$.
If an orange ball is drawn, it \emph{is replaced with another orange ball}. So the total remains 5 balls.
But the number of orange balls becomes 3 (because the drawn orange ball is replaced with another orange).
Thus the new composition is: 3 orange, 3 green? No — green was 3 originally and remains unchanged, so it is 3 green and 3 orange?
Actually, initial was 3G + 2O. One O is drawn, and replaced by a new O, so:
3G + 2O → draw O → replace with O → still 3G + 2O.
So the box composition does not change. It remains 3 green and 2 orange (5 balls total).
Thus probability of orange in next draw = $\frac{2}{5}$.
Contribution of this case:
\[ \frac{2}{5} \times \frac{2}{5} = \frac{4}{25} \]
Final probability:
\[ \frac{3}{10} + \frac{4}{25} \]
Take LCM 50:
\[ \frac{3}{10} = \frac{15}{50}, \quad \frac{4}{25} = \frac{8}{50} \]
\[ \frac{15}{50} + \frac{8}{50} = \frac{23}{50} \]
Thus, the probability of getting an orange ball in the next draw is $\frac{23}{50}$.