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Probability

Question
CBSEENMA12033609
Wired Faculty App

A black and a red dice are rolled. Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.

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: a total of 8
therefore space space straight E thin space colon space space open curly brackets left parenthesis 2 comma space 6 right parenthesis comma space left parenthesis 3 comma space 5 right parenthesis comma space left parenthesis 4 comma space 4 right parenthesis comma space left parenthesis 5 comma space 3 right parenthesis comma space left parenthesis 6 comma space 2 right parenthesis close curly brackets space space space space space space straight F thin space colon space red space die space resulted space in space straight a space number space less space than space 4 therefore space space straight F space equals space open curly brackets left parenthesis 1 comma space 1 right parenthesis comma space left parenthesis 1 comma space 2 right parenthesis comma.... comma space left parenthesis 1 comma space 6 right parenthesis comma space left parenthesis 2 comma space 1 right parenthesis comma space left parenthesis 2 comma space 2 right parenthesis comma space.... comma space left parenthesis 2 comma space 6 right parenthesis comma space left parenthesis 3 comma space 1 right parenthesis comma space left parenthesis 3 comma space 2 right parenthesis comma.... left parenthesis 3 comma space 6 right parenthesis space close curly brackets therefore space space straight E intersection straight F space equals space open curly brackets left parenthesis 2 comma space 6 right parenthesis comma space left parenthesis 3 comma space 5 right parenthesis close curly brackets therefore space space space straight P left parenthesis straight F right parenthesis space equals space 18 over 36 comma space space space straight P left parenthesis straight E intersection straight F right parenthesis space equals space 2 over 36
therefore space space required space probability space equals space 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 equals space fraction numerator begin display style 2 over 36 end style over denominator begin display style 18 over 36 end style end fraction equals space 2 over 18 space equals 1 over 9

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