Problem 7 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

A box containing 4 white and 2 black pens. Another box contains 3 white and 5 black pens. If 1 pen is selected from each box, then the probability that both the pens are white is equal to (a) \(\frac{1}{2}\) (b) \(\frac{1}{3}\) (c) \(\frac{1}{4}\) (d) \(\frac{1}{5}\)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{4} \), which corresponds to option (c).

Step 1: Determine Total Outcomes in Each Box

First, calculate the total number of pens in each box. The first box has 4 white pens and 2 black pens, making a total of 6 pens. The second box contains 3 white pens and 5 black pens, making a total of 8 pens.

Step 2: Calculate Probability from First Box

The probability of drawing a white pen from the first box is the number of white pens divided by the total number of pens. This is \( \frac{4}{6} = \frac{2}{3} \).

Step 3: Calculate Probability from Second Box

Similarly, the probability of drawing a white pen from the second box is the number of white pens divided by the total number of pens in the second box. This is \( \frac{3}{8} \).

Step 4: Calculate Combined Probability

To find the probability that both pens drawn are white, multiply the probability of drawing a white pen from the first box by the probability of drawing a white pen from the second box. Thus, the probability is \( \frac{2}{3} \times \frac{3}{8} = \frac{6}{24} = \frac{1}{4} \).

Step 5: Choose the Correct Answer

The calculated probability of both pens being white corresponds to option (c) \( \frac{1}{4} \).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics focused on counting, arrangement, and combination of elements within a set. In this exercise, we explored how to use combinatorial methods to determine the number of possible outcomes when drawing pens from two boxes. First, we had to calculate the total number of pens in each box. This was straightforward, utilizing basic counting principles. With the first box containing 6 pens (4 white and 2 black) and the second box containing 8 pens (3 white and 5 black), the exercise demonstrates how to partition a problem into smaller parts to simplify the task. Understanding the concept of permutations and combinations can further aid in complex scenarios where order or specific grouping is considered. However, here we focused more on basic probabilities derived from straightforward counting of potential outcomes. The essence of this exercise is the ability to determine individual event probabilities, leading into further exploration of combined events through conditional probability.

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already occurred. In the context of this exercise, we examined the probability of drawing a white pen from both boxes.To find the probability that both pens drawn are white, we viewed each box as a separate event. For a conditional probability scenario, the result from the first box does not affect the result from the second since they are independent events. For the first box, the probability of choosing a white pen was calculated as \( \frac{2}{3} \). Correspondingly, from the second box, the probability was \( \frac{3}{8} \).When dealing with multiple independent events, the probabilities can be multiplied to get the combined probability of both events occurring. Therefore, in this problem, since the boxes operate independently, the probability of both pens being white becomes the product of the two individual probabilities, calculated as \( \frac{2}{3} \times \frac{3}{8} = \frac{1}{4} \). This understanding is vital in various real-world situations where multiple independent conditions influence the overall outcome.

Mathematics Problem-Solving

Mathematics problem-solving is the process of understanding and solving mathematical problems in a systematic way. This exercise illustrated key problem-solving strategies that can be applied broadly in mathematics. The process started with breaking down the problem into manageable components. Initially, we identified the total number of items (pens) in each category (box), which represents a crucial problem-solving step: identifying known information. Next, calculating probabilities required understanding the relationships between individual components, showcasing the importance of step-by-step logical reasoning. This was followed by combining individual probabilities to solve for the overall probability of both pens being white. Overall, mathematics problem-solving involves strategies such as identifying given information, understanding relationships, organizing data effectively, and logically reasoning through each step. This exercise is a classic example that reinforces these fundamental skills, essential for tackling more complex mathematics problems in the future.