# 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 doesnÂ´t hear the alarm the probability is 0.5.

Please view link to see my working out:
https://ibb.co/NSCLMWc
(edited 10 months ago)
Original post by As.1997
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 doesnÂ´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.
(edited 10 months ago)
Original post by mqb2766
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!
Original post by As.1997
Thank you so much! I really liked the table method!

Its really just the joint probability which is something thats possibly a bit underemphasised at "a level".