GATE CS 2022 Q8 — General Aptitude

We are given a box containing 5 balls of the same size and shape:

• Green balls ($G$) = $3$
• Orange balls ($O$) = $2$

We want to find the probability of drawing an orange ball on the second draw ($O_2$). We can analyze this using the law of total probability based on the outcome of the first draw:

Case 1: The first ball drawn is Green ($G_1$)

• The probability of drawing a green ball first is: $$P(G_1) = \frac{3}{5}$$
• Since a green ball is drawn, it is not replaced. The box now contains $2$ green balls and $2$ orange balls (total of $4$ balls).
• The conditional probability of drawing an orange ball second is: $$P(O_2 \mid G_1) = \frac{2}{4} = \frac{1}{2}$$

Case 2: The first ball drawn is Orange ($O_1$)

• The probability of drawing an orange ball first is: $$P(O_1) = \frac{2}{5}$$
• Since an orange ball is drawn, it is replaced with another orange ball. The box still contains $3$ green balls and $2$ orange balls (total of $5$ balls).
• The conditional probability of drawing an orange ball second is: $$P(O_2 \mid O_1) = \frac{2}{5}$$

Total Probability:

Using the law of total probability, the probability of getting an orange ball in the second draw is:

Thus, the correct option is D.