Question
A bag contains 4 red, 5 blue, and 6 green balls. Three balls are drawn at random without replacement. What is the probability that exactly two are blue?
Solution — Step by Step
Total balls = 15. Total ways:
Exactly 2 blue means: choose 2 blue from 5, and 1 non-blue from the remaining 10 (4 red + 6 green).
Final answer:
Why This Works
When draws are without replacement and order doesn’t matter, combinations (not permutations) count outcomes correctly. The formula relies on every combination being equally likely.
The “exactly 2 blue” requirement means we deliberately avoid the third blue — that’s why we draw the third from the non-blue pool.
Alternative Method
Use the multiplication rule on ordered draws and then divide by the number of orderings. P(BBN) for one specific order = . There are orderings of where the non-blue lands. Multiply: . Same answer.
Common Mistake
Students compute “at least 2 blue” instead of “exactly 2 blue”. “At least 2” includes the case of all 3 blue, which adds favourable outcomes. The two answers differ — read the question carefully.
For “without replacement, count by combinations” is the default approach in CBSE. For “with replacement” or “ordered”, use multiplication of independent probabilities.