Refer to given Exercise. A psychologist has noticed that "Teachers" and "Rationalists" get along particularly well with each other, and she thinks that they tend to marry each other. One of her colleagues disagrees and thinks that the types of spouses are independent of each other.

a. If the types are independent, what is the probability that a randomly selected opposite-sex married couple would consist of one "Rationalist" and one "Teacher"?

b. In surveys of thousands of randomly selected married couples, the psychologist has found that about 5% of them have one "Rationalist" and one "Teacher." Does this contradict her colleague's theory that the types of spouses are independent of each other? Explain.

Exercise
A psychological test identifies people as being one of eight types. For instance, Type 1 is "Rationalist" and applies to 15% of men and 8% of women. Type 2 is "Teacher" and applies to 12% of men and 14% of women. Each person fits one and only one type.

a. What is the probability that a randomly selected male is either a "Rationalist" or a "Teacher"?

b. What is the probability that a randomly selected female is not a "Teacher"?

c. Suppose that college roommates have a particularly hard time getting along with each other if they are both "Rationalists." A college randomly assigns roommates of the same sex. What proportion of male roommate pairs will have this problem?

d. Refer to part (c). What proportion of female roommate pairs will have this problem?

e. Using your answers to parts (c) and (d) and assuming that half of college roommate pairs are male and half are female, what proportion of all roommate pairs will both be "Rationalists"?