Question

The supplier to the department store imports three brands of DVD players. 45% of the DVD players are Brand X, 25% are brand Y and 30% are brand Z. It is known that some of the DVD players develop a fault while under warranty. Past experience has shown that 5% of brand X, 20% of brand Y and 8% of brand Z develop a fault while under warranty.

Q1. Find the probability that a randomly chosen DVD player will develop a fault while under warranty.
2. Given that a DVD player did develop a fault while under warranty, what is the probability that it was a brand X DVD player?

Answer 1

So you have events \(X, Y, Z\) which represent the event that a random DVD is brand x, y, z respectively.

Answer 2

You \(F\) the event of a fault

And finally, they give you probabilities for \(F|X, F|Y,F|Z\), which are the conditional probabilities.

Answer 3

We know: \[
\Pr(F|X) = \frac{\Pr(F\cap X)}{\Pr(X)}
\]Which means: \[
\Pr(F\cap X) = \Pr(F|X) \cdot \Pr(X)
\]

Answer 4

If we select a DVD we know that the following is true: \[
\Pr(X\cup Y\cup Z) =1
\]

Answer 5

Also the events are mutually exclusive so: \[
\Pr(X)+\Pr(Y)+\Pr(Z)=1
\]Which means we can use total probability here:\[
\begin{split}
\Pr(F) &= \Pr(F\cap X)+\Pr(F\cap Y)+\Pr(F\cap Z)\\
&= \Pr(F|X)\cdot \Pr(X)+\Pr(F|Y)\cdot\Pr(Y)+\Pr(F|Z)\cdot\Pr(Z)
\end{split}
\]

Answer 6

really helpful, thanks