Binomial distribution — New n and p

Question

Let .

(i) Find .

(ii) Show . Hence,

(iii) Using a normal approximation and , estimate as a one-standard-deviation style statement (nearest empirical-rule percentage).

This practice question was inspired by Question 12(b) in the 2020 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.(i) .
Finish the calculation, then check that the result meets the question’s conditions.(ii) , so .
State the final answer clearly in the form the question requested.Using part (ii), (iii) to is roughly when (about ), so approximately .

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

Verification: Mean , variance are exact; the percentage uses the empirical rule.

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