Proability question

In order to write an exam, a student needs an alarm clock to wake up, which has proven to successfully wake him 80% of the time. If he hears the alarm in the morning the probability of writing the test is 0.9 and if he doesn´t hear the alarm the probability is 0.5.

Please view link to see my working out:
https://ibb.co/XYvdNyj

From the defintion of conditional probability, the joint gives bayes formula as you rearrange
p(A&B) = p(A|B)p(B) = p(B|A)p(A)
so the numerator is the joint probability which you cant simply replace with the marginal p(A).

As a 2*2 table, you could imagine that you view the problem as wrote exam (rows) and heard alarm (columns). Assuming you had 100 people the joint data would be
..................... Heard | Not Heard
Wrote........... 72.......|.....10
Not Wrote..... 8.......|......10
so each cell represent the counts assoicated with joint probability p(A & B). If we know they wrote the exam, were interested in row 1 (condition on it or slice the data) and the probaiblity they heard the alarm is 72/82, so 72/(72+10).

If we just used the marginal p(heard) on the numerator that would be 80/82, wed be including the 8 who heard but did not write the exam in the calculation (80=72+8) which doesnt really make sense. If we rewrote (modify the data sllightly) the second row as 18 0 (so leave the first row unchanged), wed end up with 90/80 which is a probability > 1 so problematic.

Rather than just remembering a formula, I generally find imagining the joint as a 2d table (in this case) makes the conditioning/given idea much more intuitive as you simply slice it on a row or column, depending on what youre given and ignore the other outcomes. Its similar to a venn diagram when you focus on one of the outcomes (circles) if youre given that that outcome occurs.

Thank you so much! I really liked the table method!