4. A training agency awards a certificate to each student who passes a test while completing a course.
Students failing the test will attempt the test again up to 3 more times, and, if they pass the test, will be awarded a certificate.
The probability of passing the test at the first attempt is 0.7 , but the probability of passing reduces by 0.2 at each attempt.

1. Complete the tree diagram below to show this information.
   \includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-08_545_1244_639_340} A student who completed the course is selected at random.
2. Find the probability that the student was awarded a certificate.
3. Given that the student was awarded a certificate, find the probability that the student passed on the first or second attempt. The training agency decides to alter the test taken by the students while completing the course, but will not allow more than 2 attempts. The agency requires the probability of passing the test at the first attempt to be \(p\), and the probability of passing the test at the second attempt to be ( \(p - 0.2\) ). The percentage of students who complete the course and are awarded a certificate is to be \(95 \%\)
4. Show that \(p\) satisfies the equation $$p ^ { 2 } - 2.2 p + 1.15 = 0$$
5. Hence find the value of \(p\), giving your answer to 3 decimal places.
   \includegraphics[max width=\textwidth, alt={}, center]{70137e9a-0a6b-48b5-8dd4-c436cb063351-09_2261_47_313_37}