Question

A bag contains different kinds of balls in which there are 5 yellow, 4 black, and 3 green balls. If 3 balls are drawn at random, then find the probability that no black ball is chosen.

• A. \( \frac{14}{55} \)
• B. \( \frac{1}{66} \)
• C. \( \frac{2}{9} \)
• D. None of these

Hint:

When calculating probabilities involving combinations, always calculate the total number of possible outcomes and the favorable outcomes, then take their ratio.

Answer 1

The Correct Option is A

Step 1: Total number of balls.
The total number of balls in the bag is: \[ 5 \, \text{(yellow)} + 4 \, \text{(black)} + 3 \, \text{(green)} = 12 \, \text{balls}. \]

Step 2: Number of ways to choose 3 balls from 12.
The total number of ways to choose 3 balls from 12 is given by the combination formula: \[ \binom{12}{3} = \frac{12 \times 11 \times 10}{3 \times 2 \times 1} = 220. \]

Step 3: Number of ways to choose 3 balls excluding black balls.
If no black ball is chosen, then we must choose 3 balls from the 5 yellow and 3 green balls (8 balls in total). The number of ways to choose 3 balls from these 8 is: \[ \binom{8}{3} = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56. \]

Step 4: Calculate the probability.
The probability that no black ball is chosen is the ratio of favorable outcomes to total outcomes: \[ P(\text{no black ball}) = \frac{\binom{8}{3}}{\binom{12}{3}} = \frac{56}{220} = \frac{14}{55}. \]

Step 5: Conclusion.
Thus, the probability that no black ball is chosen is \( \frac{14}{55} \), and the correct answer is (a).