FSMQ probability events

[Goose]

Events A and B are such that P(A or B) = 0.8, P(A) = 0.5 and P(A given B) = 0.25. Find P(A and B')

This is what I tried:
Knowing that
P(A and B) = P(A) + P(B) - P(A or B)
and knowing that
P(A given B) = P(A and B) ÷ P(B)

I subbed in P(A), P(A or B) and P(A given B) to their respective formulas:

P(A and B) = 0.5 + P(B) - 0.8
0.25 = P(A and B) ÷ P(B)

Embedding P(A and B) into the second formula:
0.25 = (0.5 + P(B) - 0.8) ÷ P(B)

Unfortunatly P(B) turns out to be a negative probability, but I need it to find out what P(A and B') is!

I would very much appreciate someone's help on this as I really want to understand probability for FSMQ.

 

[pka]

Recall that \(\mathcal{P}(A)=\mathcal{P}(A\cap B)+\mathcal{P}(A\cap B^c).\)
Work with that & see what falls out.

 

[Dr.Peterson]

Events A and B are such that P(A or B) = 0.8, P(A) = 0.5 and P(A given B) = 0.25. Find P(A and B')

This is what I tried:
Knowing that
P(A and B) = P(A) + P(B) - P(A or B)
and knowing that
P(A given B) = P(A and B) ÷ P(B)

I subbed in P(A), P(A or B) and P(A given B) to their respective formulas:

P(A and B) = 0.5 + P(B) - 0.8
0.25 = P(A and B) ÷ P(B)

Embedding P(A and B) into the second formula:
0.25 = (0.5 + P(B) - 0.8) ÷ P(B)

Unfortunatly P(B) turns out to be a negative probability, but I need it to find out what P(A and B') is!

I would very much appreciate someone's help on this as I really want to understand probability for FSMQ.

So you're solving the equation 0.25 = (b - 0.3)/b ? Please show your work, because I get a reasonable answer. I suspect a sign error.

 

[pka]

Events A and B are such that P(A or B) = 0.8, P(A) = 0.5 and P(A given B) = 0.25. Find P(A and B')

I had hoped that my reply would prompt an exploration of properties.
You should know that:
\(\mathcal{P}(A|B) = 0.25 \Rightarrow \mathcal{P}(A \cap B) = 0.25\mathcal{P}(B)\)
By you own calculations:
\(\mathcal{P}(A\cup B)=\mathcal{P}(A)+\mathcal{P}(B)-\mathcal{P}(A\cap B)\) so you get
\(0.80=0.50+\mathcal{P}(B)-0.25\mathcal{P} (B)\)
Can you find \(\mathcal{P} (B)~?\)