2006 AJC P2 [Modified]

During an online mathematics test, multiple-choice questions are randomly selected from a large item bank. The probability that a selected question is on algebra is \(\frac{4}{5}\) and that it is on geometry is \(\frac{1}{5}\). Each question has four options, among which only one is correct. A candidate Amy, who has studied algebra but not geometry, answers each algebra question correctly with a probability of \(\frac{7}{8}\). When faced with a geometry question, she simply makes a random guess. Four marks are awarded for each question answered correctly and one mark deducted for each question answered wrongly.

1. Consider a question attempted by Amy. Show that the probability that she answered the question correctly is \(\frac{3}{4}\).
2. The random variable \(X\) is the score obtained by Amy for a question. Write down the probability distribution of \(X\).
3. Assume that Amy answers \(50\) questions independently.
   If Amy obtained at least \(130\) marks, find the least number of questions that she answers correctly. Hence find the probability that Amy will obtain at least \(130\) marks.