How to use Bayes Theorem by tree diagram Question

An investment company analyses stocks and predicts whether their price will go up or down. So
  far, half of the stocks analysed by the company have gone up, $3/4$ of the stocks that went up were
  correctly predicted to go up, and $2/5$ of the stocks that went down were incorrectly predicted to
  go up. Suppose that the company tells you that it will go up. Compute the probability that the
  stock will indeed go up.

My Answer
I have drawn a tree diagram for this question based on the stocks gone up and gone down. For "gone up", $1/2$ stocks. For "gone down", $1/2$ stocks. After drawing "gone up" and "down", I also drew "correctly go up" and "incorrectly go down" for the stocks that have gone up. I also draw "correctly go down" and "incorrectly go down" for the stocks that have gone down. 
My answer for this question will be $1/2 \cdot 3/4$. It seems not correct since I didn't use the Bayes' Theorem. Can I have some tips on this question. I think I have some misunderstanding for this question 
 A: Let $N$ be the number of stocks the company analyzed. We are told that $\frac N2$ stocks went up. Of those $3/4$ were predicted to go up, and $1/4$ were not predicted to go up. That means:


*

*$\dfrac 34\dfrac N2 = \dfrac {3N}8$ stocks were predicted to go up and did go up.

*$\dfrac 14\dfrac N2 = \dfrac N8$ stocks were not predicted to go up, but went up.


The other $\frac N2$ stocks did not go up. Of these $\frac 25$ were predicted to go up. So we also have


*

*$\dfrac 25\dfrac N2 = \dfrac N5$ stocks were predicted to go up but did not go up.

*$\dfrac 35\dfrac N2 = \dfrac {3N}{10}$ stocks were not predicted to go up, and did not go up.


You have a stock that was predicted to rise. That means it is either one of the $\frac {3N}8$ stocks predicted to rise which did rise. Or it is one of the $\frac N5$ stocks predicted to rise but did not rise. The total number of stocks predicted to rise is $$\dfrac {3N}8 + \dfrac N5 = \frac{23N}{40}$$
The probability of such a stock actually rising is the ratio
$$\dfrac{\dfrac {3N}8}{\phantom{M}\dfrac{23N}{40}\phantom{M}} = \dfrac {15}{23}$$
This is also what Bayes Theorem tells you, but I find it better to work things out carefully until you understand what is going on, and only when you are comfortable with that, to resort to applying memorized formulas.
