Doubt in a 'probability problem

Question

A lot of 50 spacing washers contains 30 washers that are thicker than the target dimension. Suppose that three washers are selected at random, without replacement, from the lot.

(a) What is the probability that all three washers are thicker than the target?

(b) What is the probability that the third washer selected is thicker than the target if the first two washers selected are thinner than the target?

(c) What is the probability that the third washer selected is thicker than the target?

I had a doubt in the part (b) of the problem.

My solution :

Let A - Event that the washer is thicker

B - Event that the washer is thinner

Sample space S = {AAA , AAB , ABA , BAA , BBB , BBA , BAB , ABB }

So the only outcome which satisfies the condition in (b) is BBA so the answer should be

$(\frac{20}{50} . \frac{19}{49} . \frac{30}{48})$

But the answer given is

$(\frac{30}{48})$

Explanation given : Given that first two washers are thinner , the number of washers remaining would be 48(without replacement) , so the probability that the third washer is thicker is $(\frac{30}{48})$

I am not quite sure if this is correct , can anyone just help me out with it.

Answer 1

Of course the fastest way to solve the problem is to consider the remaining lot of 48 spacing washers 30 of which are thicker but your solution is not totally wrong.

They are requesting to calculate $P(A|B)$ and you correctly calculated $P(A\cap B)$. If you go on in your calculation you will find

$$\frac{\frac{20}{50}\frac{19}{49}\frac{30}{48}}{\frac{20}{50}\frac{19}{49}}=\frac{30}{48}$$

Answer 2

I agree with the answer 30/48. I thought about it the same way as the "correct answer". If the first two were thinner, there are 48 washing machines left to choose from for the third space. Of these 48 machines, 30 are thicker. Therefore, the probability to choose a thicker one is 30/48. Your answer answers the question what is the probability of choosing two thinner washing machines and then a thicker washing machine (part c). However the question fixes the fact that the first two were thinner. So using your expression, it would be (1 * 1* (30/48))