Question 909323: a room has 3 lamps from the collection of 10 light bulbs of which 6 are not good, a person selects 3 at random and puts them into sockets. what is the probability that, he will have light from all the three lamps.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! on your first draw, the probability of getting a good bulb is 4/10.
on your second draw, the probability of getting a good bulb is 3/9.
on your third draw, the probability of getting a good bulb is 2/8.
the probability that you will get a good bulb on all 3 draws is therefore equal to 4/10 * 3/9 * 2/8 which is equal to 24/720 which can be simplified to 1/30 which has a decimal equivalent of .033 rounded to 3 decimal places.
you can also look at it as the number of possible ways you can get a good bulb divided by the total possible ways you can get a bulb.
the equation for that is c(4,3) / c(10,3) which is equal to 4 / 120 which is equal to 1/30 again.
note that c(4,3) and 4c3 mean the same thing.
i mention this now because i used the ncx terminology down below when i used the c(n,x) terminology here.
c(n,x) is the combination formula of n! / (x! * (n-x)!)
ncx is also the combination formula of n! / (x! * (n-x)!)
when n = 10 and x = 3, this formula becomes 10! / (3! * 7!) which becomes 10*9*8*7! divided by 3! * 7! which becomes 10*9*8 / 3! which becomes 10*9*8 divided by 3*2*1 which becomes 10*3*4 which becomes 120.
similar machinations for when n = 4 and x = 3 results in 4.
you have 2 formula that both point to the same conclusion.
this indicates a fair chance that the solution is good unless you completely misunderstood what the problem is asking you to do.
this can be confirmed with a simple example that the formula is good.
assume 4 bulbs of which 3 are good and 1 is bad.
you want to draw 2 bulbs.
what is the probability that both bulbs will be good.
the formula says that the probability will be 3/4 * 2/3 = 6/12 = 1/2.
the combination formula says that the probability will be (3c2) / (4c2) which is equal to 3/6 which is equal to 1/2.
since the numbers are small, we can see what the possible combinations are:
you have bulbs labeled a, b, c, and d.
a and b and c are good and d is bad.
the total possible of combinations of 2 bulbs from the set of 4 are:
ab
ac
ad
bc
bd
cd
of these possible combinations, ad and bd and cd contain bad bulb d.
that means that you have 3 good sets of 2 and 3 bad sets of 2.
that's a probability of 1/2 that you will get 2 good bulbs from this set of 4.
the logic appears sound and should be able to be applied to the larger problem which is what i did.
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