Question

4. Abox in a certain supply room contains four 40-W light bulbs, five 60-W bulbs, and six 75-W bulbs. Suppose that two bulbs are randomly selected without replacement. (a) Given that at least one of them is found to be rated 75-W, what is the probability that both of them are 75-W bulbs? (4 points) (b) Given that at least one of them is not rated 75-W, what is the probability that both selected bulls have the same rating? (4 points)

a , b two questions please

Answer 1

solution:

Given that

No.of 40-W light bulbs = 4

No.of 60-w light bulbs = 5

No.of 75 -w light bulbs = 6

Total No.of bulbs = 4+5+6 = 15

In the event of selecting bulbs, we have

No.of ways to select 2 bulbs =n(S) = 15C2 = 105

a) Let A = event that atleast one of them is found to be rated 75-w(1 or Both)

n(A) = (6C1* 9C1) + 6C2 = 69

P(getting atleast one of them is found to be rated 75-w) = P(A)

= n(A)/n(S)  

= 69 / 105

Let B = event that Both of them is found to be rated 75-w

n(B) = 6C2 = 15

   P(getting Both of them is found to be rated 75-w) = P(B)

= n(B)/n(S)  

= 15 / 105

Here, B $\cap$ A = B [ since B is subset of A]

  $\therefore$ P(Both of selected bulbs are 75-w given that atleast one of them is 75-w ) = P(B|A)

  = P( B $\cap$ A) / P(A) [ using Bayes theorem]

= P(B) / P(A)

= (15/105) / (69/105)

= 15/ 69

b) Let A' = event that atleast one of them not rated 75-w

Here P(A') = 1 - P(getting atleast one of them is found to be rated 75-w) [ since, complement rule]

= 1 - P(A)

= 1 - 69/105

= 36/105

Let C = event that both selected have same rating

n(C) = Both are 40-w (or) Both are 60-w (or) Both are 75w

= 4C2 + 5C2 + 6C2

= 6 + 10 + 15

= 31

Here C $\cap$ A' = same rating $\cap$ atleast one of them not rated 75-w = rated 40-w (or) rated 60-w

n(C $\cap$ A' ) = 4C2 + 5C2 = 6+10 = 16

P(C $\cap$ A' ) = 16/105

$\therefore$ P(Both of selected bulbs have same rating given that atleast one of them is not rated 75-w ) = P(C|A')

  = P( C $\cap$ A') / P(A') [ using Bayes theorem]

= (16/105) / (36/105)

= 16/36

= 4/9