Phrases Previous Year Questions (PYQs) – Probability – Page 1 of 6

Probability — No Black Ball is Chosen

Given:

• Yellow balls = 5
• Black balls = 4
• Green balls = 3
• Total balls = 5 + 4 + 3 = 12

We are to find:

Probability that no black ball is selected when 3 balls are drawn at random.

Step 1: Total number of ways to choose any 3 balls from 12:

$$\text{Total ways}=(\genfrac{}{}{0ex}{}{12}{3})=220$$

Step 2: Ways to choose 3 balls such that no black ball is chosen:

Only yellow and green balls are allowed ⇒ Total = 5 (yellow) + 3 (green) = 8
$$\text{Favorable ways}=(\genfrac{}{}{0ex}{}{8}{3})=56$$

Step 3: Probability

$$P(\text{no black ball})=\frac{\text{Favorable outcomes}}{\text{Total outcomes}}=\frac{56}{220}=\frac{14}{55}$$

$$\boxed{{\displaystyle \text{Probability}=\frac{14}{55}}}$$

Conditional Probability – Bayes' Theorem

Given: One ball is transferred from Bag I to Bag II, and then a ball is drawn from Bag II and is black.

Goal: Find the probability that the transferred ball was red, given that a black ball was drawn.

Using Bayes' theorem: $$P(R|A)=\frac{P(R\cap A)}{P(A)}=\frac{\frac{3}{10}\cdot \frac{5}{10}}{\frac{3}{10}\cdot \frac{5}{10}+\frac{4}{10}\cdot \frac{6}{10}+\frac{3}{10}\cdot \frac{5}{10}}=\frac{15}{54}=\boxed{{\displaystyle \frac{5}{18}}}$$

✅ Final Answer: $\boxed{{\displaystyle \frac{5}{18}}}$