2023 TJC P2 Q6

A multiple choice test consists of \(30\) questions, with each question having \(4\) choices of which only \(1\) is correct. Each correct answer gains \(3\) marks but each incorrect answer loses \(1\) mark.

Charlie’s strategy for this test is to answer all \(30\) questions. For each question, he will choose his answer at random with each choice having equal chance of being chosen. Let \(C\) denote the number of correct answers that Charlie achieves.

1. State, in context, what must be assumed for \(C\) to be well modelled by a binomial distribution.[1]

Assume now that \(C\) has the binomial distribution \(\mathrm{B}\left( 30,0.25 \right)\).

2. Find the probability that Charlie answers at most \(10\) questions correctly.[1]
3. Show that the expected number of marks that Charlie can achieve is zero and find the variance of Charlie’s mark[4]

The passing mark for this test is \(32\).

4. Find the probability that Charlie will pass the test.[2]