John and Mary exercise. Is it correct? Here an exercise from a book about probability

John and Mary are taking a mathematics course. The course has only
  three grades: A, B, and C. The probability that John gets a B is .3.
  The probability that Mary gets a B is .4. The probability that neither
  gets an A but at least one gets a B is .1. What is the probability
  that at least one gets a B but neither gets a C?

Let's define probabilities as J(a) - John gets A, J(b), J(c), M(a), and so on.
J(b) = 0.3
M(b) = 0.4

The probability that neither gets an A but at least one gets a B is .1

means
J(b)*M(b) + J(b)*M(c) + J(c)*M(b) = 0.1

i.e.
0.3 * 0.4 + 0.3 * M(c) + J(c) * 0.4 = .1

so
0.3 * M(c) + J(c) * 0.4 = .1 - 0.12 = -.02

how can it be? where is my mistake?
 A: The probability that neither gets an A but at least one gets a B is given by:
$$
p(\text{no A's, at least one B})=p(J=B,M=B)+p(J=B,M=C)+p(J=C,M=B)
$$
You are assuming that the probabilities of John's and Mary's grades are independent. Under this assumption, your initial equation is correct, since in that case: 
$$p(J=B,M=B)=p(J=B)p(M=B)$$
$$p(J=B,M=C)=p(J=B)p(M=C)$$
$$p(J=C,M=B)=p(J=C)p(M=B)$$
and therefore $p(\text{no A's, at least one B})$ would simply be the sum of these probabilities (as it was in your equations). However, the problem description doesn't actually tell us whether John and Mary's grades are independently distributed. It might well be (for example) that John and Mary's grades are correlated, so that they tend to get similar grades. In that case, the three equalities above do not necessarily hold. The fact that, under the assumption of independence, you couldn't get a reasonable answer, suggests that the assumption of independence is indeed violated in this case.
My advice would be to write out the table of possible outcomes, and then mark the sections of this table for which you know the total probability. E.g. you know that the row of all outcomes where John gets a B has a total probability of 0.3, and so forth. If you do this for all the given probabilities, it should be obvious how to get from these to the probability that is asked for. 
