Compare two normal height models and calculate mixture and Bayes probabilities.

In a large school, the heights of all fourteen-year-old students are measured. The heights of the girls are normally distributed with mean 155 cm and standard deviation 10 cm. The heights of the boys are normally distributed with mean 160 cm and standard deviation 12 cm.

1. Find the probability that a girl is taller than 170 cm.[3]
2. Given that 10% of the girls are shorter than x cm, find x.[3]
3. Given that 90% of the boys have heights between q cm and r cm where q and r are symmetrical about 160 cm, and q < r, find the value of q and of r. In the group of fourteen-year-old students, 60% are girls and 40% are boys. The probability that a girl is taller than 170 cm was found in part (a). The probability that a boy is taller than 170 cm is 0.202. A fourteen-year-old student is selected at random.[4]
4. Calculate the probability that the student is taller than 170 cm.[4]
5. Given that the student is taller than 170 cm, what is the probability the student is a girl?[3]