Q16:
12 defective pens are accidentally mixed with 132 good ones. It is not possible to just look at a pen and tell whether or not it is defective. One pen is taken out at random from this lot. Determine the probability that the pen taken out is a good one.

Solution :

Given:

Number of defective pens = $12$

Number of good pens = $132$

To Find:

The probability that a pen taken out at random is a good one.

Step 1: Determine the total number of outcomes.

Let $n(D)$ be the number of defective pens and $n(G)$ be the number of good pens.

Total number of pens in the lot, denoted by $n(S)$, is the sum of defective and good pens.

$n(S) = n(D) + n(G)$

$n(S) = 12 + 132$

$n(S) = 144$

[Since the total number of possible outcomes is the sum of all individual items in the sample space]

Step 2: Define the event and identify favorable outcomes.

Let $E$ be the event of drawing a good pen.

The number of favorable outcomes for event $E$, denoted by $n(E)$, is equal to the number of good pens.

$n(E) = 132$

Step 3: Apply the Probability Formula.

The probability of an event $P(E)$ is defined as the ratio of the number of favorable outcomes to the total number of possible outcomes.

$P(E) = \frac{n(E)}{n(S)}$

$P(E) = \frac{132}{144}$

Step 4: Simplify the fraction.

To simplify $\frac{132}{144}$, we find the greatest common divisor (GCD) of $132$ and $144$.

Divide both numerator and denominator by their common factors:

Divide by $12$:

$132 \div 12 = 11$

$144 \div 12 = 12$

Therefore, $P(E) = \frac{11}{12}$

Final Answer: The probability that the pen taken out is a good one is $\frac{11}{12}$.

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