Gce Model Questions Vol1 2020 Further Mathematics Question 36

A company manufactures widgets in three factories X, Y and Z. The proportion of total production from X, Y and Z is 50 %, 30 % and 20 % respectively. The proportions of defective widgets from X, Y and Z are 1 %, 3 % and 4 % respectively. Each widget is subjected to two independent quality tests. Test A reports a defective widget as defective with probability \(0.9\) and a non‑defective widget as non‑defective with probability \(0.95\). Test B reports a defective widget as defective with probability \(0.8\) and a non‑defective widget as non‑defective with probability \(0.98\). A randomly selected widget passes both tests (i.e. both tests report it as non‑defective). What is the probability that the widget is actually non‑defective? Give your answer as a decimal correct to four significant figures.