February/March 2026 Paper 52

Drivers who wish to obtain an advanced driving qualification must pass two tests: a theory test and a skills test. Drivers must pass the theory test first and they are allowed up to a maximum of two attempts to pass the theory test. The probability that any driver passes the theory test on their first attempt is 0.7 and the probability that any driver passes on their second attempt is 0.6. Drivers who are successful in the theory test are then allowed up to a maximum of three attempts to pass the skills test. The probability that a driver passes the skills test on any attempt is 0.2. Complete the tree diagram to illustrate this information, showing all the probabilities.

Show that the probability that a randomly chosen driver passes the theory test at the first attempt and the skills test at the second attempt is 0.112.

Find the probability that a randomly chosen driver obtains the advanced driving qualification.

Find the probability that a randomly chosen driver passes the theory test at the first attempt given that the driver obtained the advanced driving qualification.