# Conditional probability

#### Romanka

##### New member
Please, help me to solve the problem
Details at a factory are tested randomly to check if they are faulty. It is known from previous experience that the probability of a
faulty detail is known to be 0.03. If a faulty detail is tested the probability of it testing faulty is 0.82. If a non-faulty detail is
tested the probability of it testing faulty is 0.06. Given that the detail was tested as not been faulty, calculate the
probability that it was faulty.

I understand that it's about conditional probability, but can't get it.
Thanks!

#### tkhunny

##### Well-known member
MHB Math Helper
Have you considered drawing a 2x2 grid and filling in the boxes?

#### Plato

##### Well-known member
MHB Math Helper
Details at a factory are tested randomly to check if they are faulty. It is known from previous experience that the probability of a faulty detail is known to be 0.03. If a faulty detail is tested the probability of it testing faulty is 0.82. If a non-faulty detail is tested the probability of it testing faulty is 0.06. Given that the detail was tested as not been faulty, calculate the probability that it was faulty.
Use $F$ for faulty and $T$ for a positive test.
From the given: $\mathcal{P}(F)=0.03,~\mathcal{P}(T|F)=0.82,~\&~ \mathcal{P}(T|F^c)=0.06$

Now you want $\mathcal{P}{(F|T^c)}$$=\dfrac{\mathcal{P}(F\cap T^c)}{\mathcal{P}(T^c)}. #### Romanka ##### New member thanks, I'll try ---------- Post added at 03:27 PM ---------- Previous post was at 03:26 PM ---------- Use F for faulty and T for a positive test. From the given: \mathcal{P}(F)=0.03,~\mathcal{P}(T|F)=0.82,~\&~ \mathcal{P}(T|F^c)=0.06 Now you want \mathcal{P}{(F|T^c)}$$=\dfrac{\mathcal{P}(F\cap T^c)}{\mathcal{P}(T^c)}$.
that's exactly what I need! thank you so much!

#### HallsofIvy

##### Well-known member
MHB Math Helper
Here's how I like to do problems like this: Imagine there are 10000 details (chosen to avoid fractions).

"It is known from previous experience that the probability of a faulty detail is known to be 0.03."
Okay, so our 10000 sample includes (0.03)(10000)= 300 faulty details and 10000- 300= 9700 that are not faulty.

"If a faulty detail is tested the probability of it testing faulty is 0.82."
Of the 300 faulty details, (0.82)(300)= 246 will test faulty, the other 54 will test not-faulty.

"If a non-faulty detail is tested the probability of it testing faulty is 0.06."
Of the 9700 non-faulty details, (.06)(9700)= 582 will test faulty. 9700- 582= 9118 will test not-faulty.

"Given that the detail was tested as not been faulty, calculate the probability that it was faulty."
There are a total of 9118+ 54= 9172 details that test non-faulty of which 54 are faulty.

thanks!

#### HallsofIvy

##### Well-known member
MHB Math Helper
By the way, "detail" doesn't seem like quite the word you want. "Detail" means a small part of something larger. I suspect this was translated from another language and you really wanted "item".

#### Romanka

##### New member
Yes, maybe. But I got the problem about "details"

#### Plato

##### Well-known member
MHB Math Helper
Yes, maybe. But I got the problem about "details"
YES. But what was language of the question?
Did you use a translation program to post it here?