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Given that E and F are events such that P(E) = 0.6, P(F) = 0.3 and P(E ∩ F) = 0.2, find P(E | F) and P(F | E).

Here P(E) = 0.6, P(F) = 0.3, P(E ∩ F) = 0.2
Now,

straight P left parenthesis straight E space left enclose space straight F end enclose right parenthesis space equals space fraction numerator straight P left parenthesis straight E space intersection space straight F right parenthesis over denominator straight P left parenthesis straight F right parenthesis end fraction space equals space fraction numerator 0.2 over denominator 0.3 end fraction space equals 2 over 3

and

straight P left parenthesis straight F space left enclose straight E right parenthesis space equals space fraction numerator straight P left parenthesis straight F intersection straight E right parenthesis over denominator straight P left parenthesis straight E right parenthesis end fraction space equals fraction numerator straight P left parenthesis straight E intersection straight F right parenthesis over denominator straight P left parenthesis straight E right parenthesis end fraction equals space fraction numerator 0.2 over denominator 0.6 end fraction equals space 2 over 6 space equals 1 third

215 Views

If $straight P left parenthesis straight A right parenthesis space equals space 6 over 11 comma space space space straight P left parenthesis straight B right parenthesis space equals space 5 over 11 space space and space straight P left parenthesis straight A space union space straight B right parenthesis space equals space 7 over 11 comma space space find space straight P left parenthesis straight A intersection straight B right parenthesis$ .

Here $straight P left parenthesis straight A right parenthesis space equals space 6 over 11 comma space space straight P left parenthesis straight B right parenthesis space equals space 5 over 11 comma space space straight P left parenthesis straight A union straight B right parenthesis space equals space 7 over 11$

Now $straight P left parenthesis straight A union straight B right parenthesis space equals space straight P left parenthesis straight A right parenthesis space plus space straight P left parenthesis straight B right parenthesis space minus space straight P left parenthesis straight A intersection straight B right parenthesis$
$rightwards double arrow space space space space space space space space space space space space space space space space space space space 7 over 11 space equals space 6 over 11 plus 5 over 11 minus space straight P left parenthesis straight A space intersection space straight B right parenthesis rightwards double arrow space space space space space space space space space space space space space space space space straight P left parenthesis straight A intersection straight B right parenthesis space equals space 6 over 11 plus 5 over 11 minus 7 over 11 space equals space fraction numerator 6 plus 5 minus 7 over denominator 11 end fraction space equals 4 over 11$

102 Views

Compute P(A|B), if P(B) = 0.5 and P(A ∩ B) = 0.32.

Here P(B) = 0.5, P(A ∩ B) = 0.32
Now

straight P left parenthesis straight A space left enclose straight B right parenthesis end enclose space equals space fraction numerator straight P left parenthesis straight A intersection straight B right parenthesis over denominator straight P left parenthesis straight B right parenthesis end fraction space equals space fraction numerator 0.32 over denominator 0.5 end fraction space equals space 32 over 100 cross times 10 over 5 space equals space 16 over 25

103 Views

Evaluate $straight P left parenthesis straight A union straight B right parenthesis comma space space if space 2 space straight P left parenthesis straight A right parenthesis space equals space straight P left parenthesis straight B right parenthesis space equals 5 over 13 space and space straight P left parenthesis straight A space left enclose straight B right parenthesis space equals 2 over 5.$

2 P(A) = P(B) = $5 over 13$ (given)
$therefore space space space space space space straight P left parenthesis straight A right parenthesis space equals space 5 over 26 comma space space space straight P left parenthesis straight B right parenthesis space equals space 5 over 13$

Also, $straight P left parenthesis straight A space left enclose straight B right parenthesis space equals space 2 over 5 space space space space space rightwards double arrow space space space space fraction numerator straight P left parenthesis straight A intersection straight B right parenthesis over denominator straight P left parenthesis straight B right parenthesis end fraction space equals space 2 over 5 space space space rightwards double arrow space space space space space fraction numerator straight P left parenthesis straight A intersection straight B right parenthesis over denominator begin display style 5 over 13 end style end fraction space equals 2 over 5$
$therefore space space space space space space space straight P left parenthesis straight A intersection straight B right parenthesis space equals space 2 over 5 cross times 5 over 13 space equals space 2 over 13$

Now, $straight P left parenthesis straight A union straight B right parenthesis space equals space straight P left parenthesis straight A right parenthesis plus straight P left parenthesis straight B right parenthesis minus straight P left parenthesis straight A intersection straight B right parenthesis$

$equals space 5 over 26 plus 5 over 13 minus 2 over 13 space equals fraction numerator 5 plus 10 minus 4 over denominator 26 end fraction space equals 11 over 26$