Answer to Question #328851 in Statistics and Probability for FAITH

Question #328851

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.


1
Expert's answer
2022-04-16T04:13:01-0400

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

P(A)=13,P(B)=25,P(A)=38.P(A) =\cfrac{1} {3} ,P(B) =\cfrac{2} {5} ,P(A) =\cfrac{3} {8}.

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

P(Aˉ)=113=23,P(Bˉ)=125=35,P(Cˉ)=138=58.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 Aˉ,Bˉ,Cˉ\bar{A}, \bar{B}, \bar{C} are independent,

P(AˉBˉCˉ)=P(Aˉ)P(Bˉ)P(Cˉ)==233558=14.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} .


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