# Math Help - how to compute this probability?

1. ## how to compute this probability?

I have taken this problem from the paper of MASTER OF COMMERCE Examination of university in India with little modifications.
If a machine is correctly set up it will produce 90% acceptable items. If it is incorrectly set up it will produce 40% acceptable items. Past 5 years experience shows 80% of the setups are correctly done. If after a certain setup
1) machine produces 3 acceptable items and 1 non-acceptable( defective ) item as the first 4 pieces, find the probability that machine is correctly setup.
2)machine produces 3 acceptable items as the first 3 pieces,find the probability that the machine is correctly setup.

I don't know what are the correct answers.So verification of these answers is needed.

2. ## Re: how to compute this probability?

Originally Posted by Vinod
I have taken this problem from the paper of MASTER OF COMMERCE Examination of university in India with little modifications.
If a machine is correctly set up it will produce 90% acceptable items. If it is incorrectly set up it will produce 40% acceptable items. Past 5 years experience shows 80% of the setups are correctly done. If after a certain setup
1) machine produces 3 acceptable items and 1 non-acceptable( defective ) item as the first 4 pieces, find the probability that machine is correctly setup.
2)machine produces 3 acceptable items as the first 3 pieces,find the probability that the machine is correctly setup.

I don't know what are the correct answers.So verification of these answers is needed.
By my calculations neither is right.

Tell us how you approached this and then we might be able to see where you went wrong.

CB

3. ## Re: how to compute this probability?

[TEX]
Originally Posted by CaptainBlack
By my calculations neither is right.

Tell us how you approached this and then we might be able to see where you went wrong.

CB
Firstly, I am giving calculations for the second answer
Let us define the events:
$A_1=$The set up was correct.
$A_2=$The set up was wrong.
E=The item is acceptable.
D=The item is not acceptable (defective)

We know $P(A_1)=$Probability that the set up was correct =0.8
$P(A_2)=$Probability that the set up was wrong =0.2
$P(E|A_1)=$Probability the item is acceptable given the information that the set up is correct =0.9
$P(E|A_2)=$=0.4

We are required to find the probability that the set up is correct given that the 3 items are acceptable, i-e
$P(A_1|E)=\frac{P(E|A_1)*P(A_1)^3}{P(E|A_1)*P(A_1)^ 3+P(E|A_2)*P(A_2)^3}$
$=\frac{0.9*(0.8)^3}{0.9*(0.8)^3+0.4*(0.2)^3}$=0.99310345

From the above calculations, i think you will know the calculations i have made for the first answer.

4. ## Re: how to compute this probability?

Originally Posted by Vinod
[TEX]
Firstly, I am giving calculations for the second answer
Let us define the events:
$A_1=$The set up was correct.
$A_2=$The set up was wrong.
E=The item is acceptable.
D=The item is not acceptable (defective)

We know $P(A_1)=$Probability that the set up was correct =0.8
$P(A_2)=$Probability that the set up was wrong =0.2
$P(E|A_1)=$Probability the item is acceptable given the information that the set up is correct =0.9
$P(E|A_2)=$=0.4

We are required to find the probability that the set up is correct given that the 3 items are acceptable, i-e
$P(A_1|E)=\frac{P(E|A_1)*P(A_1)^3}{P(E|A_1)*P(A_1)^ 3+P(E|A_2)*P(A_2)^3}$
$=\frac{0.9*(0.8)^3}{0.9*(0.8)^3+0.4*(0.2)^3}$=0.99310345

From the above calculations, i think you will know the calculations i have made for the first answer.
The numerator should be :

$P(\text{3 acceptable}|\text{setup OK})P(\text{setup OK})=0.9^3\times 0.8$

and the denominator:

$P(\text{3 acceptable}|\text{setup OK})P(\text{setup OK})+P(\text{3 acceptable}|\text{setup NOT OK})P(\text{setup NOT OK})\\ \phantom{xxxxxxxx}=0.9^3\times 0.8+0.4^3\times 0.2$

CB

5. ## Re: how to compute this probability?

Originally Posted by CaptainBlack
The numerator should be :

$P(\text{3 acceptable}|\text{setup OK})P(\text{setup OK})=0.9^3\times 0.8$

and the denominator:

$P(\text{3 acceptable}|\text{setup OK})P(\text{setup OK})+P(\text{3 acceptable}|\text{setup NOT OK})P(\text{setup NOT OK})\\ \phantom{xxxxxxxx}=0.9^3\times 0.8+0.4^3\times 0.2$

CB
As my answer to the second question has been corrected, my answer to the first question is

$\frac{0.046656}{0.1192}$ Verification of answer is needed.