one Student is tested in 2 tests,the probability from him to pass the 1st test is 65%

hasanneo

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Hello can you please help me build a probability table for the following problem
i have to advance in probability and i'm stuck in this frustrating question.


one Student is tested in 2 tests,the probability from him to pass the first test is 65%.and the probability for him to pass both tests is 55%,and the probability for him to pass at least one test is 80%.


I've figured out so far from the question:
let A be the probability to pass the first test and let B be the probability to pass the second test
p(A)=0.65
p(A')=0.35
P(A ∩ B)=0.55
P(A ∩ B')+P(A' ∩ B)=80%.
need directions on how to find P(A' ∩ B) and P(A' ∩ B')
 
thought:
Shouldn't the last one be:

P(A ∩ B')+P(A' ∩ B)+P(A
B)=80%

Logically, at least one test means, either the first one or the second one or both

(disclosure: school was 20 years ago...)
 
I would think it would depend on whether or not the tests are independent events. In other words, does the student's score on test A in any way influence his score on test B? If the two are indeed independent, then you can calculate the probability of passing test B with a bit of intuition. You know the probability of him passing test A is 65% and you know the probability of him passing both A and B is 55%. What then must be the probability of him passing test B? From there, you should be able to calculate the rest of the values needed to populate the table.

As a hint, you might consider a standard deck of cards. The probability of drawing a Jack is 1 in 13. The probability of drawing the Jack of Spades is 1 in 52. What then, is the odds of drawing a Spade?
 
one Student is tested in 2 tests,the probability from him to pass the first test is 65%.and the probability for him to pass both tests is 55%,and the probability for him to pass at least one test is 80%.
Using your notation:
\(\displaystyle \begin{align*} \mathscr{P}(A\cup B) &=\mathscr{P}(A)+\mathscr{P}( B)-\mathscr{P}(A\cap B)\\ 0.80 &=0.65+\mathscr{P}( B)-0.55\end{align*}\)

Now solve!
 
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