A bag contains only Pink and Green balls, with a total of 16 balls. What is the number of Pink balls in the bag?

Given statements:

(A) The probability of drawing a Pink ball, $P(P)$, is less than $0.1875$.

(B) The probability of drawing two Green balls consecutively, $P(G)$, is less than $1 - P(P)$.

(C) The difference between the number of Pink balls and Green balls is 12.

Which of the following statements is/are sufficient to find the number of Pink balls?

Question

A bag contains only Pink and Green balls, with a total of 16 balls. What is the number of Pink balls in the bag?

Given statements:

(A) The probability of drawing a Pink ball, $P(P)$, is less than $0.1875$.

(B) The probability of drawing two Green balls consecutively, $P(G)$, is less than $1 - P(P)$.

(C) The difference between the number of Pink balls and Green balls is 12.

Which of the following statements is/are sufficient to find the number of Pink balls?

Accepted Answer

**Solution:**

1. Let the number of Pink balls be $P$ and Green balls be $G$. Given total balls:
   $$P + G = 16$$

2. From statement (A):
   - Probability of drawing a Pink ball is less than $0.1875$.
   - Probability $P(P) = \frac{P}{16} < 0.1875$
   - Multiply both sides by 16:
     $$P < 3$$
   - So, $P$ can be either 1 or 2.

3. Statement (B) says the probability of drawing two Green balls consecutively is less than $1 - P(P)$.
   - This statement alone does not provide a clear numeric constraint to determine $P$ or $G$.

4. Statement (C) says the difference between the number of Pink and Green balls is 12:
   $$|P - G| = 12$$
   - Since $P + G = 16$, possible pairs are:
     - If $P - G = 12$, then $P + G = 16$ and $P - G = 12$.
       Adding both equations:
       $$2P = 28 ightarrow P = 14, G = 2$$
     - If $G - P = 12$, then $P + G = 16$ and $G - P = 12$.
       Adding both equations:
       $$2G = 28 ightarrow G = 14, P = 2$$

5. Using statement (A) and (C) together:
   - From (A), $P < 3$, so $P$ can be 1 or 2.
   - From (C), possible values for $P$ are 14 or 2.
   - Combining both, $P = 2$.

6. Therefore, statements (A) and (C) together are sufficient to find the number of Pink balls.

**Answer:** Both statements (A) and (C) together are required.

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