Two normal models for egg masses: a tail probability, a percentile, a symmetric central interval and conditional probabilities for the combined flock

A farm keeps hens of two breeds, \(A\) and \(B\). The masses of eggs laid by breed \(A\) hens are normally distributed with mean \(58\) grams and standard deviation \(4.5\) grams. The masses of eggs laid by breed \(B\) hens are normally distributed with mean \(63\) grams and standard deviation \(5\) grams.

1. Find the probability that an egg laid by a breed \(A\) hen has a mass greater than \(66\) grams.[3]
2. Of the eggs laid by breed \(A\) hens, \(15\%\) have a mass less than \(w\) grams. Find the value of \(w\).[3]
3. Of the eggs laid by breed \(B\) hens, \(80\%\) have a mass between \(q\) grams and \(r\) grams, where \(q<r\) and the interval is symmetrical about \(63\) grams. Find the value of \(q\) and the value of \(r\).[4]

Of all the eggs collected on the farm, \(70\%\) are laid by breed \(A\) hens and \(30\%\) by breed \(B\) hens. The probability that an egg laid by a breed \(B\) hen has a mass greater than \(66\) grams is \(0.274\). One egg is chosen at random from all the eggs collected.

4. Find the probability that the egg has a mass greater than \(66\) grams.[4]
5. Given that the egg has a mass greater than \(66\) grams, find the probability that it was laid by a breed \(A\) hen.[3]