Problem 165 - A Complete Resource Book in Mathematics for JEE Main

Question

An urn contains nine balls of which three are red, four are blue and two are green. The experiment is to draw three balls at random without replacement from the urn. The probability that the three balls have different color is (A) \(\frac{2}{7}\) (B) \(\frac{1}{21}\) (C) \(\frac{2}{23}\) (D) \(\frac{1}{3}\)

Step-by-Step Solution

Answer

The probability is \( \frac{2}{7} \).

Step 1: Calculate Total Combinations

First, calculate the total number of ways to draw three balls from the urn. The urn has 9 balls in total, so the number of ways to choose 3 balls from these 9 is given by the combination formula \( \binom{n}{k} \), which represents the number of ways to choose \(k\) items from \(n\) items without regard to order.\[ \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

Step 2: Calculate Favorable Combinations

Now, calculate the number of ways to choose one ball of each color. We have 3 red balls, 4 blue balls, and 2 green balls. \(1\) ball can be chosen from the red balls in \( \binom{3}{1} = 3 \) ways, \(1\) ball from the blue balls in \( \binom{4}{1} = 4 \) ways, and \(1\) ball from the green balls in \( \binom{2}{1} = 2 \) ways. Therefore, the number of favorable outcomes is the product of these combinations: \[ 3 \times 4 \times 2 = 24 \]

Step 3: Calculate the Probability

The probability of drawing three balls of different colors is the ratio of the favorable outcomes to the total outcomes. Thus, the probability is: \[ \frac{24}{84} = \frac{2}{7} \] Hence, the probability that the three balls drawn have different colors is \( \frac{2}{7} \).

Key Concepts

Combinatorics

Combinatorics is a key area in mathematics that focuses on counting the ways in which things can be arranged or combined. This topic is crucial when dealing with problems that require us to understand how objects can be selected or arranged in specific orders or groups. In the context of drawing balls from an urn, combinatorics helps us calculate various possible scenarios efficiently.

To solve such problems, we often use techniques like permutations and combinations.

• Permutations involve arrangements where the order matters, while
• Combinations typically deal with selections where order does not matter.

In our exercise, determining the total number of ways to select balls from the urn is a combinatorial task. By using the combination formula, we can precisely determine how many groups of balls can be drawn without considering the sequence in which they are selected.

Combinations

Combinations are a fundamental concept in combinatorics that allow us to figure out how a subset can be chosen from a larger set, disregarding the order of selection. This is particularly useful when calculating probabilities, as it helps simplify complex counting processes by focusing solely on the selection of items.

For example, in the "urn problem," we needed to determine how we could draw three balls from a total of nine. Combinations were the tool we used here since the order of drawing did not matter:\[ \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \] This tells us there are 84 different ways to choose any 3 balls out of a total of 9. Similarly, when figuring out how to select one ball from each color group, combinations allowed us:

• Select 1 red ball from 3,
• Select 1 blue ball from 4, and
• Select 1 green ball from 2.

Utilizing combinations simplifies such calculations, making them more manageable and straightforward.

Probability Calculations

Probability calculations involve figuring out how likely an event is to happen. In problems like drawing balls from an urn, we use probability to predict the chance of drawing a specific set of balls. Probability is calculated as the ratio of the number of favorable outcomes to the total number of possible outcomes.

In the given problem, after calculating the combinations, we found two critical numbers:

• Total possible outcomes for drawing three balls: 84
• Favorable outcomes for drawing three differently colored balls: 24

The probability of this event (drawing three differently colored balls) is therefore:\[ \frac{24}{84} = \frac{2}{7} \] This means that if we carry out the draw many times, approximately 2 out of every 7 draws will result in balls of different colors. Probability calculations like these provide a powerful way to predict outcomes and make informed decisions based on statistical expectations.