Answer to Question #171453 in Statistics and Probability for Sara

Question

https://www.chegg.com/homework-help/questions-and-answers/circuit-made-5-independent-components-shown-figure--chance-component-working-properly-094--q59663024

Answer 1

The circuit below is made of 5 independent components as shown in the figure below. 

$PC_1)=P(C_2)=0.94$

$PC_3)=0.97$

$P(C_4)=P(C_5)=085$

The events $C_1$ and $C_2$ are independent

$$P(C_1\cap C_2)=P(C_1)P(C_2)$$

The events $C_4$ and $C_5$ are independent

$$P(C_4\cap C_5)=P(C_4)P(C_5)$$

$$P(C_1\cup C_2)=P(C_1)+P(C_2)-P(C_1\cap C_2)$$

$$=P(C_1)+P(C_2)-P(C_1)P(C_2)$$

$$=0.94+0.94-0.94(0.94)=0.9964$$

$$P(C_4\cup C_5)=P(C_4)+P(C_5)-P(C_4\cap C_5)$$

$$=P(C_4)+P(C_5)-P(C_4)P(C_5)$$

$$=0.85+0.85-0.85(0.85)=0.9775$$

All five components are independent

$$P(work)=P((C_1\cup C_2)\cap C_3\cap (C_4\cup C_5))$$

$$=0.9964(0.97)(0.9775)=0.94476157$$