Trouble understanding Bayes Theorem I was watching a video on YouTube and i am not sure if the given solution is correct. Can someone confirm?

 A: Seems correct, except for the typo noted in my Comment. Let $D$ indicate 'has disease' and $T$ indicate 'tests positive'.
Bayes' Theorem states the following (denoting intersection of events as 'multiplication'):
$$P(D|T) = \frac{P(DT)}{P(T)} =
\frac{P(D)P(T|D)}{P(DT)+P(D^cT)}
= \frac{P(D)P(T|D)}{P(D)P(T|D)+P(D^c)P(T|D^c)}.$$
Sometimes, $P(T) = P(D)P(T|D)+P(D^c)P(T|D^c)$ is called the Law of Total Probability.
You are given that $P(D) = 0.01,\, P(T|D) = P(T^c|D^c) = 0.99.$ 
Then by the Complement Rule, $P(D^c) = 0.99,\, P(T|D^c) = 0.01.$ 
Plug in these numbers to get the answer claimed.
To understand these probabilities, notice that $P(DT), P(D|T),$ and $P(T|D)$ all contain the same events, but they refer to three different populations:
the first to the population of everyone who may take the test, the second to
the population of those who tested positive, and the third to the population of those who have the STD. 
Note: You can find more detailed discussions of this kind of situation in several
of the links under 'Related' in the right margin. Also, you may want to look
at Wikipedia articles on "Bayes' Theorem" and "screening test."
A: Yes even without consulting the equations it is possible to work it out from the information. See below.

