7 Each question on a multiple-choice examination paper has \(n\) possible responses, only one of which is correct. Joni takes the paper and has probability \(p\), where \(0 < p < 1\), of knowing the correct response to any question, independently of any other. If she knows the correct response she will choose it, otherwise she will choose randomly from the \(n\) possibilities. The events \(K\) and \(A\) are 'Joni knows the correct response' and 'Joni answers correctly' respectively.

1. Show that \(\mathrm { P } ( A ) = \frac { q + n p } { n }\), where \(q = 1 - p\).
2. Find \(P ( K \mid A )\). A paper with 100 questions has \(n = 4\) and \(p = 0.5\). Each correct response scores 1 and each incorrect response scores - 1 .
3. (a) Joni answers all the questions on the paper and scores 40 . How many questions did she answer correctly?
   (b) By finding the distribution of the number of correct answers, or otherwise, find the probability that Joni scores at least 40 on the paper using her strategy.