# HELP - Probability Trees!

• Dec 2nd 2006, 12:12 PM
c00ky
HELP - Probability Trees!
Hi Guys, I have the following problem. Could anybody point explain to me how to solve this problem using probability trees?

Q) Companies A, B and C are producing nails. 30%, 45% and 25% of these nails are produced by companies A, B and C respectively. Quality checks reveal that 0.2%, 0.1% and 0.15% of this type of nails produced by companies A, B and C, respectively are defectives.

i) Construct a tree diagram for this question.

ii) What is the probability that a randomly chosen nail is produced by company A and is defective?

iii) What is the probability that a randomly chosen nail is defective?

iv) What is the probability that a randomly chosen nail is defective given that the nail is produced by company A?

v) What is the probability that a randomly chosen nail is produced by company A given that the nail is defective?
• Dec 2nd 2006, 01:40 PM
CaptainBlack
Quote:

Originally Posted by c00ky
Hi Guys, I have the following problem. Could anybody point explain to me how to solve this problem using probability trees?

Q) Companies A, B and C are producing nails. 30%, 45% and 25% of these nails are produced by companies A, B and C respectively. Quality checks reveal that 0.2%, 0.1% and 0.15% of this type of nails produced by companies A, B and C, respectively are defectives.

i) Construct a tree diagram for this question.

ii) What is the probability that a randomly chosen nail is produced by company A and is defective?

iii) What is the probability that a randomly chosen nail is defective?

iv) What is the probability that a randomly chosen nail is defective given that the nail is produced by company A?

v) What is the probability that a randomly chosen nail is produced by company A given that the nail is defective?

First construct the contingency (probability) tree for the problem (see attachment).

Then your questions can be answered by looking at the terminal leaf probabilities.

RonL
• Dec 2nd 2006, 03:06 PM
c00ky
i) Construct a tree diagram for this question.
I did this OK

ii) What is the probability that a randomly chosen nail is produced by company A and is defective?
I did this OK

iii) What is the probability that a randomly chosen nail is defective?

iv) What is the probability that a randomly chosen nail is defective given that the nail is produced by company A?
I did this OK

v) What is the probability that a randomly chosen nail is produced by company A given that the nail is defective?
• Dec 2nd 2006, 05:29 PM
Soroban
Hello, c00ky!

Quote:

iii) What is the probability that a randomly chosen nail is defective?

. . $\displaystyle \begin{array}{ccc}P(A \land\text{d{e}f}) \:= & (0.3)(0.002) & =\:0.0006\\P(B \land\text{d{e}f}) \:=& (0.45)(0.001) & =\: 0.00045\\ P(C \land\text{d{e}f}) \:= & (0.25)(0.0015) & =\: 0.000375 \end{array}$

Therefore: .$\displaystyle P(\text{d{e}f}) \;=\;0.0006 + 0.00045 + 0.000375 \;=\;0.001425$

Quote:

v) What is the probability that a nail is from company A, given that it is defective?

Bayes' Theorem: .$\displaystyle P(A|\text{d{e}f}) \;=\;\frac{P(A \land\text{d{e}f})}{P(\text{d{e}f})} \;=\;\frac{0.0006}{0.001425} \:\approx\:0.42105$

• Dec 3rd 2006, 01:27 AM
c00ky
Thank you very much guys.:)