2024 RI P2 Q9

The masses of a randomly chosen bolt and a randomly chosen nut are denoted by \(M\) grams and \(W\) grams respectively. \(M\) and \(W\) are independent random variables with the distributions \(\mathrm{N}(272, 8^2)\) and \(\mathrm{N}(98, 5^2)\) respectively.

Bolts are manufactured to fit into holes in steel plates. The bolts have diameters, in cm, that follow the distribution \(\mathrm{N}(2.65, 0.03^2)\) and the diameters of the holes, in cm, follow the distribution \(\mathrm{N}(2.72, 0.02^2)\). A manufacturer sells boxes of twenty pairs of these bolts and steel plates, where each pair consists of one randomly selected bolt and one randomly selected steel plate. A pair is acceptable only if the diameter of the hole in the steel plate is at least \(0.02\) cm larger, but no more than \(0.15\) cm larger, than the diameter of the bolt. Use a normal distribution to estimate the probability that the average number of acceptable pairs in \(50\) boxes is more than \(18\).[4]