Problem 5 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

A bag contains 3 red and 7 black balls, 2 balls are taken out at random, without replacement. If the first ball taken out is red, then what is the probability that the second taken out ball is also red? (a) \(1 / 10\) (b) \(1 / 15\) (c) \(3 / 10\) (d) \(2 / 21\)

Step-by-Step Solution

Answer

The probability is \( \frac{2}{9} \).

Step 1: Understanding the Problem

We have a bag with 3 red balls and 7 black balls, making a total of 10 balls. We are asked to find the probability that the second ball drawn is red, given that the first ball drawn is red.

Step 2: Initial Setup

Let event R1 be drawing a red ball first, and event R2 be drawing a red ball second. We need to find the probability of R2 given R1, which is written as \( P(R2 | R1) \).

Step 3: Determine First Outcome

The probability that the first ball drawn is red \( P(R1) \) is calculated as the ratio of red balls to total balls: \( \frac{3}{10} \).

Step 4: Condition After First Draw

If the first ball drawn is red, there are now 2 red balls and 7 black balls left, leading to a new total of 9 balls.

Step 5: Calculate Conditional Probability

With the first ball being red, the probability that the second ball is also red is the number of remaining red balls over the total balls left: \( P(R2 | R1) = \frac{2}{9} \).

Key Concepts

Probability Theory

Probability theory is a branch of mathematics that focuses on quantifying the likelihood of various outcomes. It's about understanding the chance of an event happening. In the context of the exercise, we're dealing with what's known as "conditional probability."

• Conditional probability measures the probability of an event occurring given that another event has already occurred. This is written as \(P(A | B)\), meaning the probability of event A given event B.
• In our exercise, we're looking at \(P(R2 | R1)\), which reads as "the probability that the second ball is red given that the first ball was red."
  
• This concept helps us answer questions where the condition changes the outcome, like in scenarios without replacements—where the outcome of the first event (drawing a red ball) affects the total possibilities for the second event.

Understanding conditional probability is crucial in many real-world situations, from predicting weather events to calculating insurance risks.

Combinatorics

Combinatorics is the mathematics of counting, and it helps us figure out how many ways we can do something. It's essential in probability because knowing the total number of possible outcomes helps us determine probabilities.

• In the exercise, we used simple counting to figure out how many red and black balls were left after the first ball was drawn.


• We originally had 3 red and 7 black balls, totaling 10. After drawing one red ball, there were only 2 red balls left. The total number of outcomes was 9 because there is no replacement.


• The concept of counting without replacement is a core part of combinatorics used in probability calculations—highlighting how each choice affects the subsequent possibilities.

Combinatorics helps us simplify problems with multiple possibilities into manageable numbers by focusing on outcomes like when determining hands in a card game or seating arrangements.

Dependent Events

Dependent events are a key concept in probability that occur when the outcome or occurrence of the first event affects the outcome of the second event. The probability changes based on the results of the prior event, which is exactly what our exercise demonstrates.

• In the exercise, after drawing one red ball, the fact that there are now 2 red balls instead of 3 directly impacts the probability of drawing another red ball.


• This dependency is reflected in how we calculated \(P(R2 | R1)\). None of the balls are replaced, causing the probabilities to shift—typical of dependent events.


• Understanding the concept of dependence helps us in fields like finance, where the result of one investment could affect the future value of another.

Recognizing and reasoning through dependent events means acknowledging how the context and conditions of previous events influence future probabilities, which is crucial in predictive modeling and decision-making.