Good day all,

I'm new to probability theory and am currently working on a problem and was looking for some feedback on my work.

The question is this: One of two components is selected at random and tested. Component 1 is faulty with probability 1/5 and Component 2 is faulty with probability 1/10. What is the probability that Component 2 was the one selected given that the selected component is faulty?

So firstly, I tried working out the probability of selecting a faulty component.

Let $F$ be the event that the selected component is faulty. Then let $A$ be the event that Component 1 is faulty and let $B$ be the event that Component 2 is faulty.

$$P(F)=P(A\cup B)$$ $$=P(A)+P(B)-P(A\cap B)$$ $$\frac{1}{2}\cdot\frac{1}{5}+\frac{1}{2}\cdot\frac{1}{10}-0$$ $$=\frac{3}{20}$$

Firstly, I've no idea whether that is correct or not, but if it is, would I then proceed to calculate the conditional probability like this:

$$P(B|F)=P(B\cap F)/P(F)$$ $$=\frac{1}{20}\cdot\frac{20}{3}$$ $$=\frac{1}{3}$$

I feel like I've really missed something. I'm not sure exactly how to calculate $P(B\cap F)$ but my intuition says that $P(B\cap F)=P(B)$ in this instance. I'm also unsure of whether my working even makes sense. Any helpful feedback would be appreciated. Thanks in advance for your patience and time!

  • $\begingroup$ Yep. Everything looks fine. As a quick sanity check-you notice that component 1 is twice as likely as component 2 to be faulty, and so it makes sense that the likelihood of a faulty component being component 2 is half of the likelihood of it being component 1. $\endgroup$ – Brent Mar 7 '15 at 2:27
  • $\begingroup$ @Brent: There are a couple of minor problems: $A$ and $B$ aren’t defined correctly, and $P(F\cap B)\ne P(B)$. (Fortunately, the correct value of $P(F\cap B)$ was used in the actual calculation.) $\endgroup$ – Brian M. Scott Mar 7 '15 at 2:29

Your description of $A$ and $B$ isn’t quite right: you want $A$ to be the event that Component $1$ is selected (which you don’t actually need), and $B$ to be the event that Component $2$ is selected.

Your calculation of $P(F)$ is fine. $P(B\cap F)$ is the probability that you picked Component $2$ and Component $2$ is faulty. The events $B$ and $F$ are independent, so this $P(B)\cdot P(F)$. We’re told that the component was picked ‘at random’, which is a sloppy way of saying that each component had probability $\frac12$ of being picked. Thus, $P(B)\cdot P(F)=\frac12\cdot\frac1{10}=\frac1{20}$. Thus, your computation used the correct value of $P(B\cap F)$ (and came to the correct final result)but contrary to what you then said, this is not $P(B)$: $P(B)$ is $\frac12$.

You could also use Bayes’ theorem for this problem:

$$P(B\mid F)=\frac{P(F\mid B)\cdot P(B)}{P(F)}\;.$$

We already know that $P(F)=\frac3{20}$ and $P(B)=\frac12$, and $P(F\mid B)$ is simply the probability that Component $2$ is faulty given that it was picked. Whether or not Component $2$ is faulty does not depend on whether it was picked, so this is simply $\frac1{10}$.

  • $\begingroup$ Many thanks for that. Makes perfect sense :) $\endgroup$ – Old mate Mar 7 '15 at 2:31
  • $\begingroup$ @Eliot: You’re very welcome. $\endgroup$ – Brian M. Scott Mar 7 '15 at 2:32

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