A box contains 100 bolts and 50 nuts. It is given that `50%` bolts and `50%` nuts are rusted. Two objects are selected from the box at random. Find the probability that either both are bolts or both are rusted.

Explanation

Total number of objects = (100 + 50) = 150.
Let S be the sample space. Then,
n(S) = number of ways of selecting 2 objects out of 150
`= ""^(150)C_(2).`
Number of rusted objects
`= (50% " of " 100) + (50% " of " 50) = (50 + 25) = 75.`
Let `E_(1) =` event of selecting 2 bolts out of 100 bolts,
and `E_(2) =` event of selecting 2 rusted objects out of 75 rusted objects.
`therefore (E_(1) nn E_(2)) =` event of selecting 2 rusted bolts out of the 50 rusted bolts
`therefore n(E_(1)) =` number of ways of selecting 2 bolts out of 100
`= ""^(100)C_(2).`
`therefore n(E_(2)) =` number of ways of selecting 2 rusted objects out of 75
`= ""^(75)C_(2).`
`therefore n(E_(1) nn E_(2)) =` number of ways of selecting 2 rusted bolts out of 50
`= ""^(50)C_(2).`
`therefore P(E_(1)) = (n(E_(1)))/(n(S)) = (""^(100)C_(2))/(""^(150)C_(2)), P(E_(2)) = (n(E_(2)))/(n(S)) = (""^(75)C_(2))/(""^(150)C_(2))`
and `P(E_(1) nn E_(2)) = (n(E_(1) nn E_(2)))/(n(S)) = (""^(50)C_(2))/(""^(150)C_(2)).`
P(selecting both bolts or both rusted objects)
`= P(E_(1) " or " E_(2)) = P(E_(1) uu E_(2))`
`= P(E_(1)) + P(E_(2)) - P(E_(1) nn E_(2))`
`= (""^(100)C_(2))/(""^(150)C_(2)) + (""^(75)C_(2))/(""^(150)C_(2)) - (""^(50)C_(2))/(""^(150)C_(2)) = ((""^(100)C_(2) + ""^(75)C_(2) - ""^(50)C_(2)))/(""^(150)C_(2))`
`= ((4950 + 2775 - 1225))/(11175) = (6500)/(11175) = 260/447 = 0.58.`
Hence, the required probability is 0.58.