A problem in the theory of probability. Where am I wrong?

Question

Probability theory

The team has two interns-analysts. Each trainee gets the correct answer 14 out of 17 times. To be more confident in an important decision, the manager decided to give the same task to both analysts at once: if both get the same answer, the manager will use it, and if the answers are different, he will choose one of them at random. To what extent does this method improve the manager's chances of making the right decision?

As I understand it, there can be two situations we need: first, a situation where two interns give an equal correct answer.

Probability=14/17 * 14/17

. And the second situation is when one trainee gives the correct answer and the other one gives the wrong answer, and after that the manager accepts the correct answer:

14/17*3/17*1/2

. But the final answer confuses me. Perhaps the factor 1/2 is superfluous, but if he receives two different answers, he can choose any of the two with a probability of 1/2.

Answer 1

In my opinion, everything is correct.
Is the correct answer known?

Answer 2

You forgot the third option, both got the wrong answer.
14/17 * 14/17 -- correct
14/17 * 3/17 * 1/2 - the first one answered correctly, the second one was wrong, they chose the correct answer
3/17*14/17*1/2 - the first one was wrong, the second right, they chose the right answer
14/17 * 3/17 * 1/2 - the first one answered correctly, the second one was wrong, they chose wrong answer 3/17 * 14/17 * 1/2 - the first
one was wrong, the second one was right, they chose not the correct answer
is 3/17 * 3/17 - both were wrong.
If I correctly wrote all the outcomes, only 17 * 17 outcomes. which are divided into groups.

Answer 3

To what extent does this method increase the manager's chances of making the right decision?

Doesn't increase at all. The chances are the same as with one intern.
The manager can toss a coin before hearing the trainees' responses. Eagle - takes the answer of the first trainee, tails - the answer of the second. If the answers turn out to be the same, then the preliminary toss of a coin will not change anything. If the answers are different - it also does not matter when to make a choice of an intern, before receiving answers or after.
Thus, we can immediately select one trainee at random and wait for his answer. With a probability of 14/17 the answer will be correct.