# Probability proof - what formulas are needed here?

## Main Question or Discussion Point

If events A and B are in the same sample space:
• .
Proove that if P(A I B') > P(A) then P(B I A) < P(B)

(where B' is the Probability of A given not B)

• .
Proove that if P(A I B) = P(A) then P(B I A) = P(B)

do we assume independence here so that P(A I B) = [P(A)*P(B)]/ P(B) = P(A) and state that since P(A n B) = P(B n A) that P(B I A) = [P(B)*P(A)] / P(A) = P(B) or is it wrong to assume independence here?

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exk
For the second proof use the fact that P(A|B)=P(A&B)/P(B) and similarly for the other one. You can't assume independence, but it is easy to see that they are using the usual definition of independence P(A&B)=P(A)P(B).

Please could you show me the steps you would take

If events A and B are in the same sample space:
• .
Proove that if P(A I B') > P(A) then P(B I A) < P(B)

(where B' is the Probability of A given not B)

...
assume
$$P(A|B')>P(A)$$
then

$$\frac{P(A\cap B')}{P(B')}>P(A)$$

$$\frac{P(B'|A)P(A)}{P(B')}>P(A)$$

$$\frac{P(B'|A)}{P(B')}>1$$

$$P(B'|A)>P(B')$$

$$1-P(B'|A)<1-P(B')$$

$$P(B|A)<P(B)$$

the other one isn't much different

Thanks soo much!!