# If a Smartphone is Defective, Which Factory Made It?

## Problem 738

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.

If a smartphone of this model is found out to be detective, what is the probability that this smartphone was manufactured in factory C?

## Hint.

Use the Bayes’ theorem.

## 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 $F_C$.

By Bayes’s rule, we have
$P(F_C \mid E) = \frac{P(F_C) \cdot P(E \mid F_C)}{P(E)}.$

Now, we compute the probabilities on the right hand side.

In the post Overall Fraction of Defective Smartphones of Three Factories, we calculated that
\begin{align*}
P(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)\\
&= 0.0455.
\end{align*}
(See the post for details.)

Factory C produces $15\%$ of the smartphones, thus $P(F_C)=0.15$.
Also, the defective rate for Factory C is $7\%$. Hence $P(E \mid F_C) = 0.07$.

Inserting these values into the formula above, we get
$P(F_C \mid E) = \frac{0.15 \cdot 0.07}{0.0455} \approx 0.2308.$

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##### If At Least One of Two Coins Lands Heads, What is the Conditional Probability that the First Coin Lands Heads?

Two fair coins are tossed. Given that at least one of them lands heads, what is the conditional probability that...

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