anonymous
  • anonymous
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
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
first one is true for sure
anonymous
  • anonymous
as is the second one switch A and B and they are identical
anonymous
  • anonymous
P(A|B)= P(A) is the definition of independence, so that is true as well

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anonymous
  • anonymous
only C is false, in fact it is rather silly
anonymous
  • anonymous
Okay thank you! Could you briefly explain why C is false and D is correct?
anonymous
  • anonymous
yes sure
anonymous
  • anonymous
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
anonymous
  • anonymous
now this is not at all the same as \[P(A|B)=P(B)\] you can see first of all that they look different
anonymous
  • anonymous
Oh I see, thank you! I was confused about that for a while thanks
anonymous
  • anonymous
yw i can give an example if you like
anonymous
  • anonymous
sure all the more to help me
anonymous
  • anonymous
are you familiar with dice? that is always a good example
anonymous
  • anonymous
yes
anonymous
  • anonymous
give me a second
anonymous
  • anonymous
alrighty
anonymous
  • anonymous
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 \[P(A|B)=\frac{P(A\cap B)}{P(B)}\]
anonymous
  • anonymous
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 \[P(B)=\frac{3}{7}\]
anonymous
  • anonymous
we could do an example with a venn diagram as well
anonymous
  • anonymous
I think I understand, Thank you so much!
anonymous
  • anonymous
yw try this put \(P(A)=.8, P(B)=.5, P(A\cap B)=.4\) show that \(A\) and \(B\) are independent

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