Law of total probability - need some clarification

Two machines, A and B, operate in some factory.

Machine A produces 10% of the factory's products.

Machine B produces 90% of the factory's products.

1% of machine A's products are flawed.

5% of machine B's products are flawed.

One product randomly selected.

what is the probability that it's flawed?

So, this is what I did:

let

let

let

Now, according to the law of total probability, we'll get that:

but the answer is:

Why were and omitted?

Re: Law of total probability - need some clarification

Quote:

Originally Posted by

**Stormey** Two machines, A and B, operate in some factory. Machine A produces 10% of the factory's products. Machine B produces 90% of the factory's products.

1% of machine A's products are flawed. 5% of machine B's products are flawed.

One product randomly selected.

what is the probability that it's flawed?

So, this is what I did:

let

let

let

the answer is:

,

**That is**, probability it is flawed given A made it times probability that A made it.

Re: Law of total probability - need some clarification

Hi Plato, thanks for the help.

Quote:

Originally Posted by

**Plato** **That is**, probability it is flawed given A made it times probability that A made it.

that is exactly my question...

I'll try to rephrase.

the probability tree looks like this:

http://i39.tinypic.com/nwy7hj.png

"probability it is flawed given A made it" (meaning P(C|A) ), if we follow the tree from its root along its branches, is

"probability that A made it" (meaning P(A) ), again, according the tree it's

so "probability it is flawed given A made it times probability that A made it" should be

Re: Law of total probability - need some clarification

Quote:

Originally Posted by

**Stormey** Hi Plato, thanks for the help.

that is exactly my question...

I'll try to rephrase.

the probability tree looks like this:

http://i39.tinypic.com/nwy7hj.png
"probability it is flawed given A made it" (meaning P(C|A) ), if we follow the tree from its root along its branches, is

"probability that A made it" (meaning P(A) ), again, according the tree it's

so "probability it is flawed given A made it times probability that A made it" should be

I really don’t understand what you are asking.

While in general I am no fan of probability trees, your's makes the answer quite clear.

In order for a part to be flawed is must come from either machine A or machine B.

Re: Law of total probability - need some clarification

Quote:

Originally Posted by

**Plato** I really don’t understand what you are asking.

While in general I am no fan of probability trees, your's makes the answer quite clear.

In order for a part to be flawed is must come from either machine A or machine B.

OK, I get it now,

I misunderstood the law of total probability, and it confused me.

Thanks.

Re: Law of total probability - need some clarification

Here's how I would do it:

Assume the machines produce 1000 products

Quote:

Machine A produces 10% of the factory's products.

so machine A produces 100 of the products.

Quote:

Machine B produces 90% of the factory's products.

And machine B produces 900.

Quote:

1% of machine A's products are flawed.

1% of 100 is 1. Machine A produces 1 flawed product.

Quote:

5% of machine B's products are flawed.

5% of 900 is 45. Machine B produces 45 flawed products.

Quote:

One product randomly selected.

what is the probability that it's flawed?

Of the 1000 products 45+ 1= 46 are flawed. The probability any one is flawed is .