Three quantities are given:

**Quantity I:** Probability that at most two students (A, B, and C) solve a question, given probabilities of not answering are 0.5, 0.4, and 0.7 respectively.

**Quantity II:** Probability of drawing at least one green ball when 3 balls are drawn randomly from a bag containing 5 green and 7 red balls.

**Quantity III:** Probability that Arun (truth-telling probability $\frac{4}{5}$) and Bhavya (truth-telling probability $\frac{6}{7}$) contradict each other when stating the same fact.

Find the values of these quantities and compare them.

Question

Three quantities are given:

**Quantity I:** Probability that at most two students (A, B, and C) solve a question, given probabilities of not answering are 0.5, 0.4, and 0.7 respectively.

**Quantity II:** Probability of drawing at least one green ball when 3 balls are drawn randomly from a bag containing 5 green and 7 red balls.

**Quantity III:** Probability that Arun (truth-telling probability $\frac{4}{5}$) and Bhavya (truth-telling probability $\frac{6}{7}$) contradict each other when stating the same fact.

Find the values of these quantities and compare them.

Accepted Answer

**Solution:**

1. **Quantity I:** Probability that at most two students solve the question.

- Given probabilities of not answering: 
  - Student A: 0.5
  - Student B: 0.4
  - Student C: 0.7

- Probability that each student answers the question:
  - A answers = 1 - 0.5 = 0.5
  - B answers = 1 - 0.4 = 0.6
  - C answers = 1 - 0.7 = 0.3

- Probability all three solve the question:
  $$0.5 \times 0.6 \times 0.3 = 0.09$$

- Probability at most two solve = 1 - Probability all three solve
  $$= 1 - 0.09 = 0.91$$

2. **Quantity II:** Probability of getting at least one green ball when 3 balls are drawn from 5 green and 7 red balls.

- Total balls = 5 + 7 = 12

- Probability of no green ball (all 3 balls are red):
  $$\frac{\binom{7}{3}}{\binom{12}{3}} = \frac{35}{220} = \frac{7}{44}$$

- Probability of at least one green ball:
  $$1 - \frac{7}{44} = \frac{37}{44} \approx 0.84$$

3. **Quantity III:** Probability Arun and Bhavya contradict each other.

- Arun speaks truth with probability $\frac{4}{5}$, lies with $\frac{1}{5}$.

- Bhavya speaks truth with probability $\frac{6}{7}$, lies with $\frac{1}{7}$.

- Contradiction occurs if one tells truth and the other lies:
  $$P = P(Arun\ truth, Bhavya\ lie) + P(Arun\ lie, Bhavya\ truth)$$
  $$= \frac{4}{5} \times \frac{1}{7} + \frac{1}{5} \times \frac{6}{7}$$
  $$= \frac{4}{35} + \frac{6}{35} = \frac{10}{35} = \frac{2}{7} \approx 0.2857$$

**Comparison:**
- Quantity I = 0.91
- Quantity II = 0.84
- Quantity III = 0.2857

Hence, **Quantity I > Quantity II > Quantity III**.

**Answer:** Quantity I > Quantity II > Quantity III

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