Question

A bag contains 5 pens and 6 pencils. If a boy selects 2 articles from the bag, then what is the probability that the selected articles will be a pen and a pencil? (1) \(\frac{2}{11}\) (2) \(\frac{3}{11}\) (3) \(\frac{6}{11}\) (4) \(\frac{5}{11}\)

Short answer

Answer: (3) \(\frac{6}{11}\)

Step-by-step solution

Calculate the total number of ways to select 2 articles from the bag, which contains 5 pens and 6 pencils.

For this, we will be using the combination formula: \(C(n, r) = \frac{n!}{r! (n-r)!}\) where n is the total number of articles and r is the number of articles to be selected. In this case, n = 5 (pens) + 6 (pencils) = 11, and r = 2. So the total number of ways to select 2 articles from the bag is: \(C(11, 2) = \frac{11!}{2! (11-2)!} = \frac{11!}{2! * 9!} =\frac{11*10}{2} = 55\) possible outcomes.

Calculate the number of ways to select 1 pen and 1 pencil from the bag.

To do this, we will count the number of ways to select 1 pen from 5 pens and 1 pencil from 6 pencils. We use the combination formula for each selection and then multiply the results: \(C(5,1) * C(6,1) = \frac{5!}{1! * (5-1)!} * \frac{6!}{1! * (6-1)!} = 5 * 6 = 30\) favorable outcomes.

Calculate the probability of selecting 1 pen and 1 pencil.

Now that we have the number of favorable outcomes and the total number of possible outcomes, we can calculate the probability: \(P(\text{selecting 1 pen and 1 pencil}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}} = \frac{30}{55} = \frac{6}{11}\) Comparing the calculated probability to the given options, the correct answer is: (3) \(\frac{6}{11}\)

Key concepts

Combination Formula

The combination formula is a fundamental tool in probability and combinatorics, often represented with the symbol 'C'. It is used to calculate the number of ways you can choose a set of items from a larger group, where the order of selection does not matter. Mathematically, the combination formula for choosing 'r' items from a set of 'n' items is written as:
c(n, r) = \(\frac{n!}{r!(n-r)!}\)
Here, the '!' symbol represents factorial notation, which we will discuss in the next section. To apply the combination formula to our exercise, we first determine the total number of items 'n', which is the sum of pens and pencils in the bag, and then the number of items 'r' we want to select.

Factorial Notation

Factorial notation is symbolized by an exclamation mark '!' and represents the product of all positive integers up to a given number. For example, the factorial of 5, written as '5!', is calculated as:
5! = 5 × 4 × 3 × 2 × 1 = 120
In probability, factorial notation is vital for calculating permutations and combinations. When you come across '0!', it is defined to be equal to 1. This is used when calculating combination or permutation for picking '0' items, as it simplifies the formulas and their computations.

Probability Calculation

Probability calculation allows us to quantify the likelihood of an event occurring. The probability (P) of an event is found using the formula:
P(Event) = \(\frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}\)
It's always a value between 0 and 1, where 0 indicates an impossibility and 1 indicates certainty. In our original problem, after finding the number of favourable and total outcomes using the combination formula, we calculate the probability of selecting one pen and one pencil from the bag.

Favourable Outcomes

Favourable outcomes in the context of probability are the specific outcomes of an experiment or event that we are interested in. For example, if we're interested in drawing a pen and a pencil from a bag, each combination of one pen and one pencil that is drawn constitutes a favourable outcome. To calculate the total number of favourable outcomes, we often use the combination formula separately for each category of the item (e.g., pens and pencils) and then multiply them if the events are independent, which was the approach in our exercise example.