Let's break down this probability problem step-by-step. We are given information about pens produced in a factory and asked to find a conditional probability.

Understanding the Probability Problem

We know two key percentages about the pens produced:

• 20% of the pens are red. This is the overall probability of picking a red pen.
• 4% of the pens are red AND defective. This is the probability of picking a pen that has both characteristics.

We need to find the probability that a pen is defective, GIVEN that we already know it is red. This is a conditional probability question.

Defining Events and Probabilities

Let's define the events:

• Let event R be that the picked pen is red.
• Let event D be that the picked pen is defective.

From the question, we are given the following probabilities:

• The probability that a pen is red is 20%.
  So, $P(R) = 20\% = \frac{20}{100} = 0.20$
• The probability that a pen is red AND defective is 4%.
  So, $P(R \text{ and } D) = 4\% = \frac{4}{100} = 0.04$

We are asked to find the probability that the pen is defective GIVEN that it is red. This is written as $P(D | R)$.

Applying the Conditional Probability Formula

The formula for conditional probability $P(A | B)$ is:

$\qquad P(A | B) = \frac{P(A \text{ and } B)}{P(B)}$

In our case, we want to find $P(D | R)$, where A is event D (defective) and B is event R (red).

So, the formula becomes:

$\qquad P(D | R) = \frac{P(D \text{ and } R)}{P(R)}$

Note that $P(D \text{ and } R)$ is the same as $P(R \text{ and } D)$, which we are given as 0.04.

Final Probability Calculation

Now, we can plug in the values we have:

$P(D | R) = \frac{P(R \text{ and } D)}{P(R)} = \frac{0.04}{0.20}$

To calculate this fraction:

$\qquad \frac{0.04}{0.20} = \frac{\frac{4}{100}}{\frac{20}{100}} = \frac{4}{100} \times \frac{100}{20} = \frac{4}{20}$

Simplifying the fraction $\frac{4}{20}$:

$\qquad \frac{4}{20} = \frac{1 \times 4}{5 \times 4} = \frac{1}{5}$

As a decimal, $\frac{1}{5} = 0.2$.

So, the probability of a pen being defective if it is red is 0.2.

This means that out of all the red pens produced, 20% of them are defective.

Revision Table: Key Concepts in Probability

Additional Information: Understanding Conditional Probability

Conditional probability is crucial in many real-world applications, from medical testing to financial risk assessment and quality control in manufacturing (like our pen factory example). It helps us update our understanding of the likelihood of an event based on new information.

Think of it this way: when we are told the pen is red, our sample space (the set of all possible outcomes) changes from all pens to only the red pens. We then calculate the probability of the pen being defective within this new, smaller sample space (the red pens). The 4% figure ($P(R \text{ and } D)$) tells us the proportion of red and defective pens out of the total pens. The 20% figure ($P(R)$) tells us the proportion of red pens out of the total pens. Dividing the first by the second gives us the proportion of red and defective pens out of just the red pens.

In our case, out of every 100 pens, 20 are red, and 4 are red and defective. If you only look at the 20 red pens, 4 of them are defective. The probability of being defective among the red ones is $4/20 = 1/5 = 0.2$.