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In a lot of 12 washing machines, there are 3 defective pieces. A person has ordered 4 washing machines. Find the probability that all the four are good.

My intuition was this

probability of a machine being undefected = 9/12 (3/4)

so probability of four machine being good should be =(3/4)^4

How is that incorrect? Can anyone please point me out?

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When you take the first machine, and it is good, you do not have anymore 12 machines to choose, but only 11! And of these, only 8 are good. Try to proceed from here. – guaraqe Apr 17 '13 at 22:22
Calculate the probability of a machine being good, then remve a good machine, calculate the probability of a machine being good, remove the machine, what is the answer for 4 machines being good? – Arjang Apr 17 '13 at 22:23
Your answer is correct if you choose a machine, see if it is good, put it back into the pool, choose again (so maybe the same one, maybe not), etc. It is also close to correct if you have a large pool that is $75\%$ good, so removing one doesn't change the composition of the pool very much. – Ross Millikan Apr 17 '13 at 22:29
up vote 1 down vote accepted

This is incorrect, keep in mind that after choosing one washing machine, it is no longer there to choose from (Juan Simões points this out in the answers).

The first one has probability $9/12$ of being good (9 good ones and 12 in total). Now we take this good one. The second one has probability $8/11$ of being good. The third one has probability $7/10$ of being good. The last one has probability $6/9$ of being good. This gives us:

$$ \frac9{12} \cdot \frac8{11} \cdot \frac7{10}\cdot \frac69 = \frac{14}{55}$$

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