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(A) $E:$ Event that the sum of the observations is $8$.$E = \{(2,6), (3,5), (4,4), (5,3), (6,2)\}$$F:$ Event that the red die resulted in a number les

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$\frac{1}{9}$

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(A) $E:$ Event that the sum of the observations is $8$.$E = \{(2,6), (3,5), (4,4), (5,3), (6,2)\}$$F:$ Event that the red die resulted in a number less than $4$.$F = \{(x, y) : y $F = \{(1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), (3,2), (3,3), (4,1), (4,2), (4,3), (5,1), (5,2), (5,3), (6,1), (6,2), (6,3)\}$Number of outcomes in $F = 18$.$E \cap F = \{(5,3), (6,2)\}$.Number of outcomes in $E \cap F = 2$.$P(F) = \frac{18}{36} = \frac{1}{2}$.$P(E \cap F) = \frac{2}{36} = \frac{1}{18}$.The conditional probability $P(E|F)$ is given by $\frac{P(E \cap F)}{P(F)}$.$P(E|F) = \frac{2/36}{18/36} = \frac{2}{18} = \frac{1}{9}$.

$A$ 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 $4$.

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