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).
Of the 20 lightbulbs in a box, 2 are defective. An inspector will select 2 lightbulbs simultaneously and at random from the box. What is the probability that neither of the lightbulbs selected will be defective?
A remote island has a unique social structure. Individuals are either "Truth-tellers" (who always speak the truth) or "Tricksters" (who always lie). You encounter three inhabitants: X, Y, and Z.
X says: "Y is a Trickster"
Y says: "Exactly one of us is a Truth-teller."
What can you definitively conclude about Z?
Consider the following statements followed by two conclusions.
Statements: 1. Some men are great. 2. Some men are wise.
Conclusions: 1. Men are either great or wise. 2. Some men are neither great nor wise. Choose the correct option: