Basic Probability Help for First Year Teacher

Solution 1:

Not only you're right, but the second question is quite weird: the question asks for $P(A\cap B)$ when in the end the final result is supposedly $P(A\cup B)$. Moreover, $A$ contains $B$, which is never pointed out; the term "overlapping" is very misleading in this case. I would throw this material away or be very suspicious about every statement it makes if I were you.

Solution 2:

I'm pretty sure its an error on the course materials. The ideas of a number being odd and a number being prime are dependent on each other (an even number cannot be prime unless it is $2$), so the rule $P(A)*P(B)=P(A\cap B)$ is not valid here (it's only valid for independent events). So: trust your instincts, you are correct.

Solution 3:

What is truly amazing to me is how $11/16$ could be thought to be an answer in the second question. The provided solution at one point even explicitly lists out the $3$ odd primes in the set $\{3, 4, 5, 6, 7, 8, 9, 10\}$, and then after getting the correct answer, proceeds to continue on and perform completely irrelevant computations.

It should immediately obvious that, for a single choice from a set of $8$ numbers, the denominator of any resulting probability cannot be $16$.

I cannot even begin to speculate as to how $0.17$ was obtained for the first question, as a decimal answer gives little insight compared to a fraction.

As a teaching tool, I would advise that for the second exercise, you actually show the students how the teacher's guide went through the solution, step by step. Then ask at each step whether the reasoning was correct. If not, why not? Where does the solution break down? What would a correct solution look like? Approach it like a critique: students will be very interested in seeing that even a publisher of a textbook can make mistakes from time to time.