The problem is as follows: We have box of the following:- {A} - 40W light bulbs, N(A) = 4- {B} - 65W light bulbs, N(B) = 5- {C} - 75W light bulbs, N(C) = 6We select 2 bulbs at random. Given that at least one of these is rated 75W, what is the probability that they are both rated 75W?Using a probability tree, I was able figure out the answer by dividing the probability of both bulbs being 75W (6/15)*(5/14) by all the cases that included a 75-W bulb including the case where both were 75 W.However, how should I do this using combinatorics?

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Jayvion Tyler · Accepted Answer

Let A=Event that both bulbs are 75W and B=Event that at least one bulb is 75W. Using Bayes Theorem one can write,  P  (  A  |  B  )  =            P      (      B          |      A      )      P      (      A      )              P      (      B      )        =            P      (      A      )              P      (      B      )      since   P  (  B  |  A  )  =  1  ie if you know both bulbs are 75W then you can be certain at least one bulb is 75W.   P  (  A  ) is straightforward, assuming drawing the bulbs are independent of one another then   P  (  A  )  =      6    15    ×      5    14    =      30    210  Now consider       B    c   = Event that neither bulb is 75W, then (again assuming independence)  P  (      B    c    )  =      9    15    ×      8    14    =      72    210  And therefore,  P  (  A  |  B  )  =            30      210              1      −              72        210              =      5    23

The problem is as follows: We have box of the following: - {A} - 40W light bulbs, N(A) = 4 - {B} -

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