Total number of balls in a bag, which contains Pink and Green balls only, is 16. What is the number of Pink balls?

Statements:

(A) Probability of drawing a pink ball $P(P)

Question

Total number of balls in a bag, which contains Pink and Green balls only, is 16. What is the number of Pink balls?

Statements:

(A) Probability of drawing a pink ball $P(P) < 0.1875$

(B) Probability of drawing two green balls $P(G) < 1 - P(P)$

(C) Difference between number of pink balls and green balls is 12.

Determine which of the following options is correct:

Accepted Answer

**Solution:**

Let the number of Pink balls be $p$ and the number of Green balls be $g$.

Given total balls:

$$p + g = 16$$

We need to find $p$.

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**Statement A:** $P(P) < 0.1875$

Probability of drawing a pink ball is:

$$P(P) = \frac{p}{16} < 0.1875$$

Calculate the upper bound for $p$:

$$p < 0.1875 	imes 16 = 3$$

Since $p$ must be an integer, possible values are $p = 0, 1, 2$ or $3$.

This does not give a unique value of $p$, so **statement A alone is insufficient**.

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**Statement B:** $P(G) < 1 - P(P)$

This is always true because:

$$P(G) = \frac{g}{16} = \frac{16 - p}{16} = 1 - \frac{p}{16} = 1 - P(P)$$

So, $P(G) < 1 - P(P)$ means:

$$\frac{16 - p}{16} < 1 - \frac{p}{16}$$

Simplify:

$$\frac{16 - p}{16} < \frac{16 - p}{16}$$

This is never true (strict inequality), so statement B is contradictory or irrelevant.

Hence, **statement B alone or with others does not help**.

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**Statement C:** Difference between number of pink and green balls is 12.

$$|p - g| = 12$$

Since $p + g = 16$, consider two cases:

1. $p - g = 12$

Adding equations:

$$p + g = 16$$

$$p - g = 12$$

Add:

$$2p = 28 \Rightarrow p = 14$$

Then $g = 16 - 14 = 2$

2. $g - p = 12$

Then $g = p + 12$

Substitute in total:

$$p + (p + 12) = 16 \Rightarrow 2p + 12 = 16 \Rightarrow 2p = 4 \Rightarrow p = 2$$

Then $g = 14$

So $p$ can be either 14 or 2.

Hence, **statement C alone is insufficient** to find unique $p$.

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**Combining statements A and C:**

From statement A:

$$p < 3$$

From statement C possible values of $p$ are 2 or 14.

Since $p < 3$, $p = 2$ is the only valid value.

Thus, combining A and C, we get a unique value of $p = 2$.

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**Conclusion:**

- Statement A alone: Insufficient
- Statement C alone: Insufficient
- Statement B alone or with others: Does not help
- Statements A and C together: Sufficient

**Answer:** (b) Both statements ‘A’ and ‘C’ are required to answer the question

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