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Let's break down the problem to solve it:

The question mentions that the number of pens sold, denoted by \( X \), follows a Poisson distribution with a mean of 20. The retailer makes a profit of Rs. 1 on each pen sold, and \( Y \) represents the net profit during a week, which is also equivalent to the number of pens sold (since each pen sold corresponds to Rs. 1 profit).

Therefore, \( Y = X \), and we need to find \( P(Y \geq 60) \), which is equivalent to \( P(X \geq 60) \).

Given \( X \sim \text{Poisson}(20) \), the probability mass function of a Poisson-distributed variable is:

\[
P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}
\]

where \( \lambda = 20 \).

Now, \( P(X \geq 60) \) can be written as:

\[
P(X \geq 60) = 1 - P(X \leq 59)
\]

The cumulative probability \( P(X \leq 59) \) can be calculated as the sum of probabilities from \( X = 0 \) to \( X = 59 \):

\[
P(X \leq 59) = \sum_{y=0}^{59} \frac{e^{-20} 20^y}{y!}
\]

Thus, the correct answer is option **(c)**:

\[
P(Y \geq 60) = 1 - \sum_{y=0}^{59} \frac{e^{-20} 20^y}{y!}
\]

Let me know if you'd like further clarification or have any questions!

Here are five related questions:
1. How do you calculate the cumulative probability for a Poisson distribution?
2. What is the significance of the parameter \( \lambda \) in a Poisson distribution?
3. How can the Poisson distribution be used in real-life scenarios?
4. How does the Poisson distribution compare to the binomial distribution?
5. Why is the sum used to find \( P(X \leq 59) \) truncated at 59?

**Tip**: The Poisson distribution is particularly useful for modeling rare events in a fixed interval of time or space.

Suppose that a retailer has 50 pens in stock and let X denote the number of pens that the retailer sells in a week. Assume that X follows the Poisson distribution with mean 20 and suppose that the retailer makes a profit of Rs. 1/- on each pen sold. Let Y denote the net profit made by the retailer during a week. The value of P(Y >= 60) equals:

Solve a Poisson distribution problem where a retailer sells pens with a mean of 20 pens per week, and calculate the probability of earning Rs. 60 or more in profit. This problem covers key concepts like Poisson probability calculation and cumulative probabilities.

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