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

To solve the problem of determining the probability of drawing an orange ball after the first draw, we need to consider the outcomes of the first draw and how they affect the probability of the subsequent draw.

1. Initially, there are 5 balls in the box: 3 green and 2 orange.
2. The probability of the event depends on what happens in the first draw. There are two cases to consider:
   • If a green ball is drawn first, the count of green balls decreases by one (since the green ball is not replaced), and the distribution becomes: 2 green and 2 orange balls. The probability of drawing an orange ball in the next draw is then \(\frac{2}{4} = \frac{1}{2}\).
   • If an orange ball is drawn first, the distribution remains: 3 green and 2 orange balls (since the orange ball is replaced). The probability of drawing an orange ball in the next draw remains \(\frac{2}{5}\).
3. We now determine the probability of each case occurring:
   • The probability of drawing a green ball first is \(\frac{3}{5}\).
   • The probability of drawing an orange ball first is \(\frac{2}{5}\). 
4. Using the law of total probability, the probability of drawing an orange ball in the next draw is calculated as:

P(\text{orange second}) = P(\text{green first}) \cdot P(\text{orange | green first}) + P(\text{orange first}) \cdot P(\text{orange | orange first})

P(\text{orange second}) = \left(\frac{3}{5} \times \frac{1}{2}\right) + \left(\frac{2}{5} \times \frac{2}{5}\right)

P(\text{orange second}) = \frac{3}{10} + \frac{4}{25} = \frac{15}{50} + \frac{8}{50} = \frac{23}{50}

1. Thus, the probability of drawing an orange ball in the next draw is \(\frac{23}{50}\).

Therefore, the correct answer is \(\frac{23}{50}\).