Question

A bag contains $40$ pen drives out of which x are non-defective. If one pen drive is drawn at random, the probability of drawing a non-defective pen drive is y. Now place this pen drive and $20$ more non-defective pen drives in this bag. Now, if a pen drive is drawn from the bag the probability of drawing the non-defective pen drive is $4y$. Find $x$.

Answer 1

Probability defines the chance of a certain event to occur. The sum of all the events that can occur in a given situation is equal to$1$. We will find the probability of drawing non defective pen drives in both the cases and equate them with the expression given to us. Hence, we will get two relations between $x$ and $y$. Solving both, we can find the value of $x$.

Complete step by step answer:
Given, total number of pen drives is $n(s) = 40$
Number of pen drives that are non-defective in the bag $= x$
As per the question, the probability of drawing a non-defective pen drive is $y = \dfrac{{{\text{Number of non defective pen drives}}}}{{{\text{Total no}}{\text{. of pen drives}}}}$
$\Rightarrow y = \dfrac{x}{{40}} - - - - (1)$
Now, $20$ more non-defective pen drives are added.
Now, the total number of pen drives in the bag becomes, $n(s) = 40 + 20$
$\Rightarrow n(s) = 60$
The number of non-defective pen drives in the bag $= x + 20$

Now, we are given that the probability of drawing a non-defective pen drive becomes$4y$.
Therefore, according to the question, $4y = \dfrac{{{\text{Number of non - defective pen drives in the bag}}}}{{{\text{Total number of pen drives in the bag}}}}$
$\Rightarrow 4y = \dfrac{{x + 20}}{{60}} - - - (2)$
So, we have two relations in x and y.
Now, substituting the value of $y$ from $(1)$ in $(2)$.
$\Rightarrow 4\left( {\dfrac{x}{{40}}} \right) = \left( {\dfrac{{x + 20}}{{60}}} \right)$
Cancelling the common factors in numerator and denominator, we get,
$\Rightarrow \dfrac{x}{{10}} = \dfrac{{x + 20}}{{60}}$

Now, cross multiplying the terms of the equation,
$\Rightarrow 60(x) = 10(x + 20)$
Dividing both sides of the equation by $10$,
$\Rightarrow 6x = x + 20$
Subtracting $x$from both sides of equation,
$\Rightarrow 6x - x = 20$
$\Rightarrow 5x = 20$
Dividing both sides of the equation by $5$, we get,
$\therefore x = 4$
So, the value of x is $4$.

Therefore, the number of non-defective pen drives is $4$.

Note: We should always remember the basic formula of probability of an event as $\left( {\dfrac{{{\text{Number of favourable outcomes}}}}{{{\text{Total number of outcomes}}}}} \right)$. We must have a clear understanding of simplification rules and transposition in order to solve the equation formed in two variables. Also, if the event has $p$ probability to occur then the probability that the event doesn’t occur is $(1 - p)$. If any event has probability equal to$1$, then such the event is said to be an obvious event. On the other hand, if an event has $0$ probability to occur, then such an event is said to be a null event.