# Overall Fraction of Defective Smartphones of Three Factories

## Problem 735

A certain model of smartphone is manufactured by three factories A, B, and C. Factories A, B, and C produce $60\%$, $25\%$, and $15\%$ of the smartphones, respectively.

Suppose that their defective rates are $5\%$, $2\%$, and $7\%$, respectively. Determine the overall fraction of defective smartphones of this model.

## Solution.

Let $E$ be the event that a smartphone of this model is defective. Let $F_A$ be the event that a smartphone is manufactured by factory A. Similarly for $F_B$ and $F_C$.

Then the overall fraction of defective smartphones of this model can be found as follows.
\begin{align*}
P(E) &= P(F_A \cap E) + P(F_B \cap E) + P(F_C \cap E)\\
&= P(F_A)\cdot P(E \mid F_A) + P(F_B)\cdot P(E \mid F_B) + P(F_C)\cdot P(E \mid F_C)\\
&= (0.6)(0.05) + (0.25)(0.02) + (0.15)(0.07)\\
&= 0.0455.
\end{align*}
Thus, the overall defective rate is $4.55\%$.

### Further Question

In the context of the above problem, if a smartphone of this model is found out to be detective, find the probability that this smartphone was manufactured in factory C.

The solution is available in the post If a Smartphone is Defective, Which Factory Made It?

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##### Complement of Independent Events are Independent

Let $E$ and $F$ be independent events. Let $F^c$ be the complement of $F$. Prove that $E$ and $F^c$ are...

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