1. [SOLVED] Card statistics

Number 1

Which are always true?
* P(SUT)=P(S)+P(T)
If S is a subset of T, P(SUT)=P(S)
P(not S)=100%-P(S)
* If S is a subset of T, P(ST)=P(S)
* If S is a subset of T, P(SUT)=P(T)

Number 2
T implies not S. Which are always true?

not S is a subset of T
not T is a subset of S
* S and T are mutually exclusive
*S and T are independent
*not T is a subset of not S

Number 3
P(S) = 90% and P(T) = 55%

Which are always true?
*P(ST)>=45%
P(SUT)>=95%
*T cannot imply S
S and T cannot be mutually exclusive
P(SUT)>=90%

* = ones I think are true

>>>>Can somebody explain which ones I got wrong? <<<

2. Hello, Starry!

* = ones I think are true

Number 1:

Which are always true?

$(a)\;\;P(A \cup B)\:=\:P(A)+P(B)$

$(b)\;\;\text{If }A \subset B,\;\text{ then }P(A \cup B)\:=\:P(A)$

$(c)\;\;P(\sim A)\:=\:100\% -P(A)$ *

$(d)\;\;\text{If }A \subset B,\;\text{ then }P(A\cap B)\:=\:P(A)$ *

$(e)\;\;\text{If }A \subset B,\:\text{ then } P(A\cap B)\:=\:P(B)$

$(f)\;\;\text{For any event }A\!:\;P(A) \,\geq\, 0$ *

$(g)\;\;P(A\cap B)\: \leq\:P(A)$ *

$(h)\;\;\text{If }A\cap B\:=\: \emptyset,\:\text{ then }P(A\cup B) \:=\: P(A) +P(B)$ *

$(i)\;\;P(A \cap B) \:=\:P(A)\!\cdot\!P(B)$

$(j)\;\;P(S)\:=\:100\%$ *

$(k)\;\;\text{If }A \subset B,\:\text{ then }P(A \cup B)\:=\:P(B)$ *

$(l)\;\;P(A \cup B) \:\leq \:P(A)+P(B)$ *

$(m)\;\;P(A \cup B) \:=\:P(A)+P(B)-P(A \cap B)$ *

All correct . . . Good work!

3. How about problems 2 and 3?

I'm sure my answers for 2 are wrong :P

4. Originally Posted by Starry
How about problems 2 and 3?
2 looks correct. What part are you not sure about?

I see a problem with 3a.
$P(A)+P(B)-P(AB)\le 1$.

5. I see...!

so because 90+55-P(ST) <1,
P(ST) would have to be greater than 45!

Thanks!
Also for problem 3, S and T cannot be mutually exclusive right?