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

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

 

 

 

 
 
 
 
 

Answers (1)

The sample space has 36 outcomes. Let A be the event that the sum of observations is 8.
\therefore A= \left \{ \left ( 2,6 \right ),\left ( 3,5 \right ),\left ( 5,3 \right ),\left ( 4,4 \right ) ,\left ( 6,2 \right )\right \}
  \Rightarrow n\left ( A \right )= 5
 \Rightarrow p\left ( A \right )= \frac{5}{36}
Let B be the event that observation on red die is less than 4.
\therefore B=\left \{ \left ( 1,1 \right ),\left ( 2,1 \right ) ,\left ( 3,1 \right ),\left ( 4,1 \right ),\left ( 5,1 \right ),\left ( 6,1 \right ),\left ( 1,2 \right ),\left ( 2,2 \right ),\left ( 3,2 \right ),\left ( 4,2 \right ),\left ( 5,2 \right ),\left ( 6,2 \right ),\left ( 1,3 \right ),\left ( 2,3 \right ),\left ( 3,3 \right ),\left ( 4,3 \right ),\left ( 5,3 \right ),\left ( 6,3 \right )\right \}
\therefore n\left ( B \right )= 18
\Rightarrow p\left ( B \right )= \frac{18}{36}= \frac{1}{2}
Clearly A\cap B= \left \{ \left ( 5,3 \right ),\left ( 6,2 \right ) \right \}
\therefore n\left ( A\cap B \right )= 2
\therefore p\left ( A\cap B \right )= \frac{2}{36}= \frac{1}{18}
p\left ( \frac{A}{B} \right )= \frac{p\left ( A\cap B \right )}{p\left ( B \right )}= \frac{\frac{1}{18}}{\frac{1}{2}}
= \frac{2}{18}= \frac{1}{9}

Posted by

Ravindra Pindel

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