Question

A black and a red dice are rolled.Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5.

Solution

Here $straight S space equals space open curly brackets table row cell left parenthesis 1 comma space 1 right parenthesis end cell cell left parenthesis 1 comma space 2 right parenthesis end cell cell left parenthesis 1 comma space 3 right parenthesis end cell cell left parenthesis 1 comma space 4 right parenthesis end cell cell left parenthesis 1 comma space 5 right parenthesis end cell cell left parenthesis 1 comma space 6 right parenthesis end cell row cell left parenthesis 2 comma space 1 right parenthesis end cell cell left parenthesis 2 comma space 2 right parenthesis end cell cell left parenthesis 2 comma space 3 right parenthesis end cell cell left parenthesis 2 comma space 4 right parenthesis end cell cell left parenthesis 2 comma space 5 right parenthesis end cell cell left parenthesis 2 comma space 6 right parenthesis end cell row cell left parenthesis 3 comma space 1 right parenthesis end cell cell left parenthesis 3 comma space 2 right parenthesis end cell cell left parenthesis 3 comma space 3 right parenthesis end cell cell left parenthesis 3 comma space 4 right parenthesis end cell cell left parenthesis 3 comma space 5 right parenthesis end cell cell left parenthesis 3 comma space 6 right parenthesis end cell row cell left parenthesis 4 comma space 1 right parenthesis end cell cell left parenthesis 4 comma space 2 right parenthesis end cell cell left parenthesis 4 comma space 3 right parenthesis end cell cell left parenthesis 4 comma space 4 right parenthesis end cell cell left parenthesis 4 comma space 5 right parenthesis end cell cell left parenthesis 4 comma space 6 right parenthesis end cell row cell left parenthesis 5 comma space 1 right parenthesis end cell cell left parenthesis 5 comma space 2 right parenthesis end cell cell left parenthesis 5 comma space 3 right parenthesis end cell cell left parenthesis 5 comma space 4 right parenthesis end cell cell left parenthesis 5 comma space 5 right parenthesis end cell cell left parenthesis 5 comma space 6 right parenthesis end cell row cell left parenthesis 6 comma space 1 right parenthesis end cell cell left parenthesis 6 comma space 2 right parenthesis end cell cell left parenthesis 6 comma space 3 right parenthesis end cell cell left parenthesis 6 comma space 4 right parenthesis end cell cell left parenthesis 6 comma space 5 right parenthesis end cell cell left parenthesis 6 comma space 6 right parenthesis end cell end table close curly brackets$
Number of elements (outcomes) of the above sample space is 6 × 6 = 36.
Let E : sum greater than 9
$therefore space space space straight E space equals space open curly brackets left parenthesis 4 comma space 6 right parenthesis comma space left parenthesis 5 comma space 5 right parenthesis comma space left parenthesis 5 comma space 6 right parenthesis comma space left parenthesis 6 comma space 4 right parenthesis comma space left parenthesis 6 comma space 5 right parenthesis comma space left parenthesis 6 comma space 6 right parenthesis close curly brackets$
Let E: black dice resultant in a 5.
$therefore space space space space straight F thin space colon thin space open curly brackets left parenthesis 5 comma space 1 right parenthesis comma space left parenthesis 5 comma space 2 right parenthesis comma space left parenthesis 5 comma space 3 right parenthesis comma space left parenthesis 5 comma space 4 right parenthesis comma space left parenthesis 5 comma space 5 right parenthesis comma space left parenthesis 5 comma space 6 right parenthesis close curly brackets$
$straight E space intersection thin space straight F space equals space open curly brackets left parenthesis 5 comma space 5 right parenthesis comma space left parenthesis 5 comma space 6 right parenthesis close curly brackets$
$therefore space space space space straight P left parenthesis straight F right parenthesis space equals space 6 over 36 comma space space space space straight P left parenthesis straight E intersection straight F right parenthesis space equals space 2 over 36$
Required probability = $straight P left parenthesis straight E left enclose straight F right parenthesis end enclose space equals space fraction numerator straight P left parenthesis straight E intersection straight F right parenthesis over denominator straight P left parenthesis straight F right parenthesis end fraction space equals fraction numerator begin display style 2 over 36 end style over denominator begin display style 6 over 36 end style end fraction space equals space 2 over 6 space equals space 1 third.$

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Probability

A black and a red dice are rolled.Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5.

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