Artificial intelligent assistant

calculating Probability of cars being faulty my question is: In 100 cars research shows 3 to be faulty. If 3 cars are selected at random, what is the probabiity that exactly 2 are faulty. So here is my working so far. I'm not too good at maths but as far as I have determined: (1/100 x 2/99 x 3/98) = 1/161700 - This is the probability of all 3 cars being faulty. I'm a bit confused so I tried multiplying that answer by 2/99 cause it says 2 are faulty. But that is giving me an 1/8004150 - This number is too big and it's not right according to the answer which is supposed to be 97/53900 I then tried reverse working it and I found by timsing by 2 get me 1/53900. I am currently trying to work backwards, but its not getting anywhere

Select the cars one by one an for $i=1,2,3$ let $F_i$ denote the event that the car chosen as number $i$ is faulty. Then the event that exactly $2$ cars will be faulty is:$$E:=(F_1\cap F_2\cap F_3^{\complement})\cup(F_1\cap F_2^{\complement}\cap F_3)\cup(F_1^{\complement}\cap F_2\cap F_3)$$

This is a union of $3$ disjoint sets and we can calculate the probability of e.g. the first like this:$$P(F_1\cap F_2\cap F_3^{\complement})=P(F_1)P(F_2\mid F_1)P(F_3^{\complement}\mid F_1\cap F_2)=\frac3{100}\frac2{99}\frac{97}{98}$$

On a similar way the two others can be calculate and it will appear that we find the same outcome.

Then we find:$$P(E)=3\times\frac3{100}\frac2{99}\frac{97}{98}$$

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Making use of the theory concerning hypergeometric distribution we find the outcome:$$\frac{\binom32\binom{97}1}{\binom{100}3}$$

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