Answer to Question #328851 in Statistics and Probability for FAITH

Question

The probability that Tope passes an exam is 1/3 and probabilities that his brother

and sister pass the same exam are 2/5 and 3/8 respectively. Find the probability that

the three of them will fail the exam.

Answer 1

Let events $A, B, C -$ Tope, his brother and his sister pass an exam.

$P(A) =\cfrac{1} {3} ,P(B) =\cfrac{2} {5} ,P(A) =\cfrac{3} {8}.$

Events that they fail an exam are $\bar{A}, \bar{B}, \bar{C}$ respectively, their probabilities

$P(\bar{A} ) =1-\cfrac{1} {3}=\cfrac{2} {3} ,\\ P(\bar{B} ) =1-\cfrac{2} {5}=\cfrac{3} {5} ,\\ P(\bar{C} ) =1-\cfrac{3} {8}=\cfrac{5} {8}.$

The events $\bar{A}, \bar{B}, \bar{C}$ are independent,

$P(\bar{A}\cap\bar{B}\cap\bar{C}) =P(\bar{A}) \cdot P(\bar{B}) \cdot P(\bar{C}) =\\ =\cfrac{2} {3} \cdot\cfrac{3} {5} \cdot\cfrac{5} {8} =\cfrac{1} {4} .$