Binomial distribution — Exactly three successes

Question

If , find .

This practice question was inspired by Question 13(d) in the 2025 NSW HSC Mathematics Extension 1 examination, © NESA. It is an original TestMum variation that practises the same mathematical idea, not an official NESA question.

Set up the solution: identify the random variable and its parameters before using the matching probability formula.
Simplify the previous line carefully, keeping exact values where possible.
Finish the calculation, then check that the result meets the question’s conditions.
State the final answer clearly in the form the question requested.Final answer:

Common mistake: changing so many features that the new example no longer tests the same mathematical idea.

Verification: The probability is between zero and one and is the largest central term.

Any mark labels are a TestMum study aid, not an official NESA marking allocation.