Q21(i):
A lot consists of 144 ball pens of which 20 are defective and the others are good. Nuri will buy a pen if it is good, but will not buy if it is defective. The shopkeeper draws one pen at random and gives it to her. What is the probability that (i) She will buy it ?

Solution :

Given:

To Find:

The probability that Nuri will buy the pen. Nuri buys the pen if and only if it is a good pen.

Step 1: Determine the number of good pens.

Let $N_{total}$ be the total number of pens and $N_{defective}$ be the number of defective pens.

The number of good pens ($N_{good}$) is calculated as:

$N_{good} = N_{total} - N_{defective}$

$N_{good} = 144 - 20$

$N_{good} = 124$

[Since the total lot consists only of good and defective pens, subtracting the defective count from the total yields the count of good pens.]

Step 2: Define the event and the probability formula.

Let $E$ be the event that Nuri buys the pen. Nuri buys the pen if it is good.

The number of favorable outcomes for event $E$ is the number of good pens, which is $124$.

The total number of possible outcomes is the total number of pens, which is $144$.

The probability of an event $P(E)$ is given by the formula:

$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Step 3: Calculate the probability.

$P(\text{She will buy it}) = \frac{124}{144}$

To simplify the fraction, we find the greatest common divisor (GCD) of $124$ and $144$.

Divide both numerator and denominator by $4$:

$124 \div 4 = 31$

$144 \div 4 = 36$

$P(E) = \frac{31}{36}$

[Since $31$ is a prime number and does not divide $36$, the fraction is in its simplest form.]

Final Answer: The probability that she will buy the pen is $\frac{31}{36}$.

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