Question

A bag contains 50 bolts and 150 nuts. Half of the bolt and half of the nuts are rusted. If an item is chosen at random, find the probability that it is rusted or that it is a bolt.
$\left( A \right)$ 3/8
$\left( B \right)$ 5/8
$\left( C \right)$ 7/8
$\left( D \right)$ None

Answer 1

Hint – In this particular question first separate the rusted bolts and nuts respectively then use the concept that probability is the ratio of the favorable number of outcomes to the total number of outcomes so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given data:
A bag contains 50 bolts and 150 nuts.
So the total number of items in the bag = (50 + 150) = 200
Now it is given that half the number of bolts and half the number of nuts are rusted.
So the rusted bolts are (50/2) = 25 bolts.
And the rusted nuts are (150/2) = 75 nuts.
So the total number of rusted bolts and the nuts are (25 + 75) = 100
Now as we know that the probability is the ratio of the favorable number of outcomes to the total number of outcomes.
Therefore, P = $\dfrac{{{\text{favorable number of outcomes}}}}{{{\text{total number of outcomes}}}}$
Now we have to find the probability that the randomly chosen item is rusted or that it is a bolt.
For rusted
Favorable number of outcomes = total rusted items = 100.
And the total number of outcomes = 200
So the probability (${P_1}$) that the randomly chosen an item is rusted is
$\Rightarrow {P_1} = \dfrac{{100}}{{200}} = \dfrac{1}{2}$
Now for a rusted bolt
Favorable number of outcomes = total rusted bolts = 25
And the total number of outcomes = 200
So the probability (${P_2}$) that the randomly chosen an item is a rusted bolt is
$\Rightarrow {P_2} = \dfrac{{25}}{{200}} = \dfrac{1}{8}$
So the probability that the randomly choose an item is rusted or that it is a bolt is the sum of both the probabilities which is calculated above,
Therefore, P = ${P_1} + {P_2}$
Now substitute the values we have,
Therefore, P = $\dfrac{1}{2} + \dfrac{1}{8} = \dfrac{{4 + 1}}{8} = \dfrac{5}{8}$
So this is the required probability.
Hence option (B) is the correct answer.

Note – Whenever we face such types of questions the key concept we have to remember is that the required probability is the sum of the probabilities of the that the randomly chosen an item is rusted and the randomly chosen an item is a rusted bolt, so first calculate these probabilities as above and then take the sum of these as above and simplify we will get the required probability.