udaymathapati wrote:
A box contains 10 light bulbs, fewer than half of which are defective. Two bulbs are to be drawn simultaneously from the box. If n of the bulbs in box are defective, what is the value of n?
(1) The probability that the two bulbs to be drawn will be defective is 1/15.
(2) The probability that one of the bulbs to be drawn will be defective and the other will not be defective is 7/15.
This little beauty should be in the 700 + category
Excellent solutions by
Bunuel and
flukeOne should be able to understand the answer by Bunuel Method easily.
FLUKE method is also equally elegant and one should follow it if they are finding it hard to understand the concept of symmetry in probability.
Some people are finding it hard to visualize the concept of symmetry in Probability
Here is an easy way to understand symmetry using two dice and their sum.
The minimum sum when two dice are thrown is 2 and maximum sum is 12.
We will see how the chances of getting sum from 2 to 12 are NOT EQUALLY LIKELY and how these chances are symmetrical in nature
1 way to get a sum of 2 = (1,1)
2 way to get a sum of 3 = (1,2)(2,1)
3 way to get a sum of 4 = (1,3) (2,2) (3,1)
4 way to get a sum of 5 = (1,4)(2,3)(3,2)(4,1)
5 way to get a sum of 6 = (1,5)(2,4)(3,3)(4,2)(5,1)
6 way to get a sum of 7 = (1,6)(2,5)(3,4)(4,3)(5,2)(6,1)
5 way to get a sum of 8 = (2,6)(3,5)(4,4)(5,3)(6,2)
4 way to get a sum of 9 = (3,6)(4,5)(5,4)(6,3)
3 way to get a sum of 10 = (4,6)(5,5)(6,4)
2 way to get a sum of 11 = (5,6)(6,11)
1 way to get a sum of 12= (6,6)
As you can see the probability of getting a sum of 6 is much higher (five times higher) when two dice are thrown than of getting a sum of 2 when two dice are thrown.
Now mentally compare it with the probability of getting 6 when only dice is thrown (Probability=1/6) and getting 2 (Probability= 1/6)
This is where symmetry comes into play. IT MAKES SOME EVENTS MORE LIKELY THEN OTHER EVENTS.
Similarly the chances of Getting one good and one bad bulb in this example are symmetrical.
2 Ways to get one good bulb and one bad bulb= (First Good, Second Bad) OR (First Bad, Second Good)
Therefore while dealing with the probability of combination of one good and one bad bulb , we should keep in mind that the the probability is counted twice
Hence
2* Probability (Good Bulb and Bad bulb)= 7/15
Hope this helps !!