Problem 10 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

A bag \(x\) contains 3 white balls and 2 black balls and another bag \(y\) contains 2 white balls and 4 black balls. A bag and a ball out of it are picked at random. The probability that the ball is white is (a) \(3 / 5\) (b) \(7 / 15\) (c) \(1 / 2\) (d) None

Step-by-Step Solution

Answer

The probability that the ball is white is \( \frac{7}{15} \) (option b).

Step 1: Understand the Problem

We have two bags, each with a different number of white and black balls. We need to find the probability that a randomly selected ball from a randomly selected bag is white.

Step 2: Determine the Probability of Choosing Each Bag

Since there are two bags, and each bag is equally likely to be chosen, the probability of choosing either bag is \( \frac{1}{2} \).

Step 3: Calculate the Probability of Picking a White Ball from Each Bag

- For bag \(x\), which has 3 white and 2 black balls, the probability of picking a white ball is \( \frac{3}{5} \).- For bag \(y\), which has 2 white and 4 black balls, the probability of picking a white ball is \( \frac{2}{6} = \frac{1}{3} \).

Step 4: Calculate the Total Probability of Picking a White Ball

Using the law of total probability:- Probability from bag \(x\): \( \frac{1}{2} \times \frac{3}{5} = \frac{3}{10} \).- Probability from bag \(y\): \( \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \).Add these probabilities together to get the total probability: \( \frac{3}{10} + \frac{1}{6} = \frac{9}{30} + \frac{5}{30} = \frac{14}{30} = \frac{7}{15} \).

Key Concepts

Law of Total Probability

The law of total probability is a fundamental concept in probability theory that allows us to find the probability of an event by considering all possible ways that event can occur. In our exercise with the bags, the event we're interested in is picking a white ball. There are two pathways for this to happen: picking a white ball from bag \(x\) or picking a white ball from bag \(y\). Each pathway has its own probability.To apply the law of total probability, we start by calculating the probability of each smaller event (in this case, selecting a specific bag and then picking a white ball). Since each bag is equally likely to be chosen, the probability of selecting either bag is \(\frac{1}{2}\). Then, we multiply this by the probability of selecting a white ball from each bag:

• For bag \(x\): the chance is \(\frac{3}{5}\), leading to a combined probability of \(\frac{1}{2} \times \frac{3}{5} = \frac{3}{10}\).
• For bag \(y\): the chance is \(\frac{1}{3}\), leading to a combined probability of \(\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\).

Finally, we add these probabilities together to get the total probability of picking a white ball, which is \(\frac{7}{15}\). This method ensures that every potential path to our desired outcome is considered.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It’s a key concept for understanding how probabilities change based on new information. In our problem, determining the probability of picking a white ball given we have already chosen a specific bag exemplifies conditional probability.For bag \(x\), the probability of picking a white ball once this bag is selected is seen as \(\frac{3}{5}\). This is because we’re only considering outcomes within bag \(x\), and the chances of drawing a white ball given you already picked bag \(x\) shifts from a global perspective to a local one.Similarly, for bag \(y\), this conditional probability is calculated to be \(\frac{1}{3}\). These probabilities illustrate how conditional probabilities adjust your likelihood of an event once some conditions are set (like choosing a bag first).Understanding how conditions affect the probability of events can be crucial in both day-to-day reasoning and complex statistical tasks. It represents a focus on specifics rather than the entire sample space.

Combinatorics

Combinatorics is the branch of mathematics dealing with the counting, arrangement, and combination of objects. Although the given problem does not explicitly call for complex combinatorial calculations, its principles underpin why we can logically break the problem into more manageable parts. Choosing a ball from one of the bags involves basic combinatorial ideas since we're dealing with combinations of different choices — one of two bags and a specific color of ball per bag. Combinatorics helps us determine the number of ways we can select items under different configurations, which is central when calculating probabilities. In our example, the configuration involves:

• Selecting one out of two bags.
• Choosing specific balls (white or black) from a selected bag with a defined combination of balls.

Thus, while direct combinatorial calculation is not utilized here, understanding how these calculations work is critical when dealing with more complex problems or when problems involve greater numbers of choices and selections. By breaking down larger problems, combinatorics simplifies our approach, making tackling probability scenarios much easier.