Question

which of the following statements are true?
A. P(A and B) = P(A) * P(B|A)
B. P(A and B) = P(B) * P(A|B)
C. P(A|B)= P(B) if A and B are independent
D. P(A|B)= P(A) if A and B are independent

Answer 1

first one is true for sure

Answer 2

as is the second one
switch A and B and they are identical

Answer 3

P(A|B)= P(A) is the definition of independence, so that is true as well

Answer 4

only C is false, in fact it is rather silly

Answer 5

Okay thank you! Could you briefly explain why C is false and D is correct?

Answer 6

yes sure

Answer 7

first of all
\[P(A|B)\] which reads "the probability of A given B" is the probability that A occurs if you know B has occurred
if knowing B has occurred gives no information about A occurring, i.e. if \[P(A|B)=P(A)\] the you say A and B are "independent"
that is the definition of independence

Answer 8

now this is not at all the same as
\[P(A|B)=P(B)\] you can see first of all that they look different

Answer 9

Oh I see, thank you! I was confused about that for a while thanks

Answer 10

yw
i can give an example if you like

Answer 11

sure all the more to help me

Answer 12

are you familiar with dice? that is always a good example

Answer 13

yes

Answer 14

give me a second

Answer 15

alrighty

Answer 16

lets do this a simpler way
suppose
\[P(A)=\frac{2}{3}\], A and B are independent so
\[P(A|B)=\frac{2}{3}\] as well
by definition
\[P(A|B)=\frac{P(A\cap B)}{P(B)}\]

Answer 17

but this does not mean that \(P(B)\) must also be \(\frac{2}{3}\) it just means the ratio must be \(\frac{2}{3}\) so for example it could be
\[\frac{P(A\cap B)}{P(B)}=\frac{\frac{2}{7}}{\frac{3}{7}}\] making
\[P(B)=\frac{3}{7}\]

Answer 18

we could do an example with a venn diagram as well

Answer 19

I think I understand, Thank you so much!

Answer 20

yw
try this
put \(P(A)=.8, P(B)=.5, P(A\cap B)=.4\)
show that \(A\) and \(B\) are independent