MHB Tree Diagram - Error at production

mathmari
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Hey! 😊

One box contains ten production parts, four of which have a production error.
Three parts are removed from the box one after the other without putting them back. We consider the following events :
A : the first production part that is removed is OK
B : the second production part that is removed is OK
C : the third production part that is removed is OK

Do we get the below diagramm?

1637242881585.png


:unsure:
 
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mathmari said:
One box contains ten production parts, four of which have a production error.
Three parts are removed from the box one after the other without putting them back. We consider the following events :
A : the first production part that is removed is OK
B : the second production part that is removed is OK
C : the third production part that is removed is OK

Do we get the below diagram?
Hey mathmari!

After we pick a good item with probability $\frac 6{10}$, there are only 9 parts left in the box, aren't there?
Shouldn't the probability on another good item then be $\frac 59$ instead of $\frac 5{10}$? (Worried)
 
Klaas van Aarsen said:
After we pick a good item with probability $\frac 6{10}$, there are only 9 parts left in the box, aren't there?
Shouldn't the probability on another good item then be $\frac 59$ instead of $\frac 5{10}$? (Worried)

Ah yes! So should it be as follows?

1637263594135.png
:unsure:
 
Looks good to me. (Nod)
 
Klaas van Aarsen said:
Looks good to me. (Nod)

Do we have the following then ?
\begin{align*}&P(A)=\frac{6}{10} \\ &P(B\mid A)=\frac{5}{9} \\ &P(A^c\cap B^c)=P(B^c\mid A^c)\cdot P(A^c)=\frac{3}{9}\cdot \frac{4}{10}=\frac{2}{15} \\ &P(B)=P(B\cap A)+P(B\cap A^c)=P(B\mid A)\cdot P(A)+P(B\mid A^c)\cdot P(A^c)=\frac{5}{9}\cdot\frac{6}{10}+\frac{6}{9}\cdot \frac{4}{10} =\frac{1}{3}+\frac{4}{15}=\frac{3}{5}\\ &P(A\mid B)=\frac{P(A\cap B)}{P(B)} =\frac{P(B\mid A)\cdot P(A)}{P(B)}=\frac{\frac{5}{9}\cdot \frac{6}{10} }{\frac{3}{5}}=\frac{\frac{1}{3} }{\frac{3}{5}} =\frac{5}{9} \\ & P(A^c\cap B^c\cap C^c) =P(C^c\mid A^c\cap B^c)\cdot P(A^c\cap B^c)=P(C^c\mid A^c\cap B^c)\cdot \frac{2}{15} =\ \ ? \\ & P(A\cup B\cup C) = \ \ ? \\ & P(C\mid A\cap B^c) = \ \ ?\end{align*}

Could you give me a hint for the last three ones? :unsure:
 
Let's consider $P(C^c\mid A^c\cap B^c)$.
It reads as: given that the first part was bad and given that the second part was also bad, what is the chance that the third part is bad?
How many parts are left in the box, and how many of them are bad?
We can also see it in your tree diagram. 🤔

According to the general sum rule we have $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. 🤔

Perhaps we can do $P(C\mid A\cap B^c)$ in the same way as $P(C^c\mid A^c\cap B^c)$. 🤔
 
Klaas van Aarsen said:
Let's consider $P(C^c\mid A^c\cap B^c)$.
It reads as: given that the first part was bad and given that the second part was also bad, what is the chance that the third part is bad?
How many parts are left in the box, and how many of them are bad?
We can also see it in your tree diagram. 🤔

Is it $\frac{2}{8}$ ? :unsure:

Can we calculate that also without using the tree diagram? :unsure:
Klaas van Aarsen said:
According to the general sum rule we have $P(A\cup B)=P(A)+P(B)-P(A\cap B)$. 🤔

\begin{align*}P(A\cup B\cup C) &= P(A\cup B)+P(C)-P(A\cap B\cap C)\\ & =P(A\cup B)+P(C)-P(A\cap B\cap C)\\ & =P(A)+P(B)-P(A\cap B)+P(C)-P(A\cap B\cap C)\\ & =P(A)+P(B)-P(A\mid B)\cdot P(B)+P(C)-P(A\cap B\cap C)\end{align*} Every term is known, without the last one, right? How can we calculate this one? :unsure:
 
mathmari said:
Is it $\frac{2}{8}$ ?

Can we calculate that also without using the tree diagram?

Yep. (Nod)

After taking 2 bad parts (given $A^c\cap B^c$), there are 2 bad parts left on a total of 8 parts. So we get $\frac 2 8$.
Isn't that how we constructed the tree diagram, and also how we found $P(A)$, $P(B\mid A)$, and so on? 🤔
mathmari said:
\begin{align*}P(A\cup B\cup C) &= P(A\cup B)+P(C)-P(A\cap B\cap C)\\ & =P(A\cup B)+P(C)-P(A\cap B\cap C)\\ & =P(A)+P(B)-P(A\cap B)+P(C)-P(A\cap B\cap C)\\ & =P(A)+P(B)-P(A\mid B)\cdot P(B)+P(C)-P(A\cap B\cap C)\end{align*} Every term is known, without the last one, right? How can we calculate this one?
I'm afraid that is not correct. (Shake)

Instead we should have $P(A\cup B\cup C) = P(A\cup B)+P(C)-P((A\cup B)\cap C)$. (Sweating)

$P(A\cap B\cap C)$ is the chance that we get 3 parts that are OK in a row. Isn't that $\frac 6{10}\cdot\frac 5{9}\cdot \frac 4{8}$? 🤔
 
So do we have the following ? \begin{align*}P(A\cup B\cup C) &= P(A\cup B)+P(C)-P((A\cup B)\cap C)\\ & =P(A\cup B)+P(C)-P((A\cap C)\cup (B\cap C))\\ & =P(A)+P(B)-P(A\cap B)+P(C)-P(A\cap C)-P(B\cap C)+P(A\cap B\cap C)\\ & =P(A)+P(B)-P(A\mid B)\cdot P(B)+P(C)-P(C\mid A)\cdot P(A)-P(C\mid B)\cdot P(B)+P(A\cap B\cap C)\\ & =\frac{6}{10}+\frac{3}{5}-\frac{5}{9}\cdot \frac{3}{5}+\left (\frac{6}{10}\cdot \frac{5}{9}\cdot \frac{4}{8}+\frac{6}{10}\cdot \frac{4}{9}\cdot \frac{4}{8}+\frac{4}{10}\cdot \frac{6}{9}\cdot \frac{5}{8}+\frac{4}{10}\cdot \frac{3}{9}\cdot \frac{6}{8}\right )-P(C\mid A)\cdot \frac{6}{10}-P(C\mid B)\cdot \frac{3}{5}+\frac{6}{10}\cdot \frac{5}{9}\cdot \frac{4}{8}\end{align*}
Is everything correct so far? :unsure:

Does it hold that $P(C\mid A)=\frac{5}{9}\cdot \frac{4}{8}+\frac{4}{9}\cdot \frac{5}{8}$ and $P(C\mid B) = \frac{4}{8}+\frac{5}{8}$ ? (The second one is wrong since it is greater than $1$.) ? :unsure:
 
  • #10
mathmari said:
So do we have the following ? \begin{align*}P(A\cup B\cup C) &= P(A\cup B)+P(C)-P((A\cup B)\cap C)\\ & =P(A\cup B)+P(C)-P((A\cap C)\cup (B\cap C))\\ & =P(A)+P(B)-P(A\cap B)+P(C)-P(A\cap C)-P(B\cap C)+P(A\cap B\cap C)\\ & =P(A)+P(B)-P(A\mid B)\cdot P(B)+P(C)-P(C\mid A)\cdot P(A)-P(C\mid B)\cdot P(B)+P(A\cap B\cap C)\\ & =\frac{6}{10}+\frac{3}{5}-\frac{5}{9}\cdot \frac{3}{5}+\left (\frac{6}{10}\cdot \frac{5}{9}\cdot \frac{4}{8}+\frac{6}{10}\cdot \frac{4}{9}\cdot \frac{4}{8}+\frac{4}{10}\cdot \frac{6}{9}\cdot \frac{5}{8}+\frac{4}{10}\cdot \frac{3}{9}\cdot \frac{6}{8}\right )-P(C\mid A)\cdot \frac{6}{10}-P(C\mid B)\cdot \frac{3}{5}+\frac{6}{10}\cdot \frac{5}{9}\cdot \frac{4}{8}\end{align*}
Is everything correct so far?

Yep. (Nod)

mathmari said:
Does it hold that $P(C\mid A)=\frac{5}{9}\cdot \frac{4}{8}+\frac{4}{9}\cdot \frac{5}{8}$ and $P(C\mid B) = \frac{4}{8}+\frac{5}{8}$ ? (The second one is wrong since it is greater than $1$.) ?
How did you get them? (Wondering)
It looks as if $P(C\mid A)$ is correct, although I needed a couple of steps to get there.

As for the other one, don't we have:
$$P(C\mid B)=\frac{P(B\cap C)}{P(B)}=\frac{P((A\cap B\cap C)\cup (A^c\cap B\cap C))}{P(B)}
=\frac{P(A\cap B\cap C)+ P(A^c\cap B\cap C)}{P(B)}$$
🤔
 
  • #11
Ahh ok!

So do we have the following ?
\begin{align*}&P(A)=\frac{6}{10} \\ &P(B)=\frac{3}{5} \\ &P(A\mid B)=\frac{5}{9} \\ &P(A\cap B\cap C) = \text{Wkt dass wir 3 Teile nacheinander nehmen die in Ordnung sind}=\frac 6{10}\cdot\frac 5{9}\cdot \frac 4{8}=\frac 1{6}
\\ & P(C)=P(C\cap B\cap A)+P(C\cap B^c\cap A)+P(C\cap B\cap A^c)+P(C\cap B^c\cap A^c)\\ & =\frac{6}{10}\cdot \frac{5}{9}\cdot \frac{4}{8}+\frac{6}{10}\cdot \frac{4}{9}\cdot \frac{4}{8}+\frac{4}{10}\cdot \frac{6}{9}\cdot \frac{5}{8}+\frac{4}{10}\cdot \frac{3}{9}\cdot \frac{6}{8} = \frac{1}{6}+ \frac{2}{15}+\frac{1}{6}+\frac{1}{10}=\frac{17}{30} \\ &P(C\mid A) =\frac{P(A\cap C)}{P(A)}=\frac{P((A\cap B\cap C)\cup (A\cap B^c\cap C))}{P(A)}
=\frac{P(A\cap B\cap C)+ P(A\cap B^c\cap C)}{P(A)}\\ & =\frac{\frac{1}{6}+ \frac{2}{15}}{\frac{6}{10}} =\frac{\frac{3}{10}}{\frac{6}{10}}=\frac{1}{2} \\ &P(C\mid B)=\frac{P(B\cap C)}{P(B)}=\frac{P((A\cap B\cap C)\cup (A^c\cap B\cap C))}{P(B)}
=\frac{P(A\cap B\cap C)+ P(A^c\cap B\cap C)}{P(B)} \\ & =\frac{\frac{1}{6}+ \frac{1}{6}}{\frac{3}{5}} =\frac{\frac{2}{6}}{\frac{3}{5}}=\frac{\frac{1}{3}}{\frac{3}{5}}=\frac{5}{9} \end{align*}
:unsure:
 
  • #12
Looks good to me! (Nod)
 
  • #13
Klaas van Aarsen said:
Perhaps we can do $P(C\mid A\cap B^c)$ in the same way as $P(C^c\mid A^c\cap B^c)$. 🤔

The term $P(C\mid A\cap B^c)$ means: given that the first part was good and given that the second part was bad, what is the chance that the third part is good?
After taking $2$ parts,one goodand one bad, (given $A\cap B^c$), there are $5$ good parts left on a total of $8$ parts. So we get $\frac 5 8$, i.e. $P(C\mid A\cap B^c)=\frac 5 8$.

Is that correct? :unsure:
 
  • #14
mathmari said:
The term $P(C\mid A\cap B^c)$ means: given that the first part was good and given that the second part was bad, what is the chance that the third part is good?
After taking $2$ parts,one goodand one bad, (given $A\cap B^c$), there are $5$ good parts left on a total of $8$ parts. So we get $\frac 5 8$, i.e. $P(C\mid A\cap B^c)=\frac 5 8$.

Is that correct?
Yep. (Nod)
 
  • #15
Klaas van Aarsen said:
Yep. (Nod)

Great! 🤩

We could calculate that also using the formula, right?
\begin{align*}P(C\mid A\cap B^c)=\frac{P(A\cap B^c\cap C)}{P(A\cap B^c)}=\frac{\frac{2}{15}}{P(B^c\mid A)\cdot P(A)}=\frac{\frac{2}{15}}{\frac{4}{9}\cdot \frac{6}{10}}=\frac{\frac{2}{15}}{ \frac{4}{15}}=\frac{2}{ 4}\end{align*}
I get a different result as previously... So can we not do that in that way? :unsure:
 
  • #16
mathmari said:
We could calculate that also using the formula, right?
\begin{align*}P(C\mid A\cap B^c)=\frac{P(A\cap B^c\cap C)}{P(A\cap B^c)}=\frac{\frac{2}{15}}{P(B^c\mid A)\cdot P(A)}=\frac{\frac{2}{15}}{\frac{4}{9}\cdot \frac{6}{10}}=\frac{\frac{2}{15}}{ \frac{4}{15}}=\frac{2}{ 4}\end{align*}
I get a different result as previously... So can we not do that in that way?

Don't we have that $P(A\cap B^c\cap C)=\frac 6{10}\cdot\frac 49\cdot \frac 58=\frac 16$ instead of $\frac 2{15}$? 🤔
 
  • #17
Klaas van Aarsen said:
Don't we have that $P(A\cap B^c\cap C)=\frac 6{10}\cdot\frac 49\cdot \frac 58=\frac 16$ instead of $\frac 2{15}$? 🤔

Ah yes, I had a typo above! (Tmi)

I want to check now if the events $B$ and $A^c$ are independent, once by calculating and once by the tree diagram.

By calculating we have :

We have that $P(A^c\cap B)=P(B\mid A^c)\cdot P(A^c)=\frac{6}{9}\cdot \frac{4}{10}= \frac{4}{15}$ and $P(A^c)\cdot P(B)=\frac{4}{10}\cdot \frac{3}{5}=\frac{12}{50}$, sowe have that $P(A^c\cap B)\neq P(A^c)\cdot P(B)$, which means that the events $A^c$ and $B$ are not independent.

Is that correct? :unsure: How can we check that at the tree diagram? :unsure:
 
  • #18
mathmari said:
I want to check now if the events $B$ and $A^c$ are independent, once by calculating and once by the tree diagram.

By calculating we have :
We have that $P(A^c\cap B)=P(B\mid A^c)\cdot P(A^c)=\frac{6}{9}\cdot \frac{4}{10}= \frac{4}{15}$ and $P(A^c)\cdot P(B)=\frac{4}{10}\cdot \frac{3}{5}=\frac{12}{50}$, sowe have that $P(A^c\cap B)\neq P(A^c)\cdot P(B)$, which means that the events $A^c$ and $B$ are not independent.

Looks correct. (Nod)

mathmari said:
How can we check that at the tree diagram?

We have 2 branches that correspond to $B$. One with $\frac 59$ and one with $\frac 69$.
Which branch we should have depends on $A$ and they do not have the same probability.
Therefore $B$ depends on $A$, which implies it also depends on $A^c$. 🤔

We can also express it as $P(B\mid A)=\frac 59\ne \frac 69=P(B\mid A^c)$, which shows that the probability on $B$ depends on $A^c$ and they are therefore not independent. 🤔
 
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