Question

12 defective pens are accidentally mixed with 132 good ones. It is not possible to tell by looking at a pen whether it is defective or not. One pen is taken out at random from this lot. What is the probability that the pen taken out is a good one?

• A. \( \frac{15}{18} \)
• B. \( \frac{13}{15} \)
• C. \( \frac{10}{12} \)
• D. \( \frac{11}{12} \)

Hint:

In probability questions: \[ P(E)=\frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} \] Always remember to calculate the total number of objects first before applying the formula.

Answer 1

The Correct Option is D

Concept: Probability is a mathematical measure of the chance of occurrence of an event. If an experiment has several possible outcomes and all outcomes are equally likely, then the probability of an event \(E\) is given by: \[ P(E)=\frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \] where:
• Favorable outcomes = outcomes that support the required event
• Total outcomes = total number of all possible outcomes The probability value always lies between 0 and 1.

Step 1: Understanding the given information carefully.
We are told that:
• Number of defective pens \(=12\)
• Number of good pens \(=132\) All the pens are mixed together. One pen is selected randomly. We have to find the probability that the selected pen is a good pen.

Step 2: Finding the total number of pens.
The total number of pens is obtained by adding the defective pens and the good pens. \[ \text{Total pens} = 12 + 132 \] \[ =144 \] Thus: \[ n(S)=144 \] where \(n(S)\) represents the total number of possible outcomes in the sample space.

Step 3: Finding the favorable outcomes.
The event required is: \[ E = \text{Selecting a good pen} \] The number of good pens is: \[ 132 \] Thus: \[ n(E)=132 \]

Step 4: Applying the probability formula.
Using: \[ P(E)=\frac{n(E)}{n(S)} \] Substituting the values: \[ P(\text{Good Pen}) = \frac{132}{144} \]

Step 5: Simplifying the fraction.
Both 132 and 144 are divisible by 12. Dividing numerator and denominator by 12: \[ \frac{132}{144} = \frac{132 \div 12}{144 \div 12} \] \[ = \frac{11}{12} \] Thus: \[ P(\text{Good Pen})=\frac{11}{12} \]

Step 6: Checking the options carefully.
Option (1): \[ \frac{15}{18} \] Incorrect. Option (2): \[ \frac{13}{15} \] Incorrect. Option (3): \[ \frac{10}{12} \] Incorrect. Option (4): \[ \frac{11}{12} \] Correct. Final Conclusion: The probability that the pen selected is a good pen is: \[ \boxed{\frac{11}{12}} \] Hence, the correct answer is option (4).