# Stuck on a probability question

1. Mar 13, 2012

### silkdigital

Question:

Dreamboat cars are produced at three different factories A, B, and C. Factory A produces 20% of the total output of Dreamboats, B 50%, and C 30%. However, 5% of the cars at A are lemons, 2% at B are lemons, 10% at C are lemons. If you buy a Dreamboat and it turns out to be a lemon, what is the probability that it was produced at factory A?

My workings were:

0.05 / (0.05 + 0.02 + 0.1) = 29.41%

I ignored the output percentages because the base condition is that the car is a lemon already. Is my logic on the right track? Not sure if I'm correct!

Another tough question is this:

Suppose that 10 cards, of which 7 are red and 3 are green, are put at random into 10 envelopes, of which seven are red and three are green, so that each envelope contains one card. Determine the probability that exactly k envelopes will contain a card with a matching color.

I've managed to obtain k=4
[7C4 * 3C3] / 10C7

and k=10
[7C7 * 3C0] / 10C7

Not sure what the next steps are to express in terms of any k. Hope that makes sense.
Thanks for any help in advance, having some difficulty!

2. Mar 13, 2012

### awkward

Silkdigital,

For the dreamboats, you need
$$P(\text{produced at A} | \text{is a lemon}) = \frac{P(\text{produced at A and is a lemon)}}{P(\text{is a lemon})}$$
for which I think you need the output percentages.

For the cards, it may be easier to answer a closely related question: How many green cards are in green envelopes? This is exactly like taking a random sample of three cards without replacement and asking how many of them are green.

3. Mar 14, 2012

### SW VandeCarr

Note, you failed to weight your error percentages by the production percentages. The weights can be assigned in various ways provided the proportions are preserved. So for factory A: (0.2)(0.05)/[(0.2)(0.05)+(0.5)(0.02)+(0.3)(0.10)] = 0.01/(0.01+0.01+0.03)=0.2

So if you have a defect, the probabilities are 0.2 from A, 0.2 from B and 0.6 from C. Note they sum to 1.

4. Mar 14, 2012

### silkdigital

Yes I realised it now. I guess I misinterpreted the question. I did the same thing and ended with 0.2, for the envelope question I realised matches only exist for even k when k>=4 (ie 4,6,8,10) and all marginal probabilities add to 1.
Thanks for the help guys!

5. Mar 14, 2012

### SW VandeCarr

You're welcome.