Understanding Conditional Probability in the Context of Ball Selection

Conditional probability is a fundamental concept in probability theory, often used in real-life scenarios such as ball selection from different boxes. This article will walk through the process of calculating conditional probability with a practical example involving two boxes of colored balls.

The Basics of Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as ( P(A|B) ), which reads 'the probability of event A given event B'. The formula for conditional probability is:

Formula for Conditional Probability

[ P(A|B) frac{P(A cap B)}{P(B)} ]

( P(A|B) ) is the conditional probability of event A given event B. ( P(A cap B) ) is the probability that both events A and B occur. ( P(B) ) is the probability of event B.

This formula helps us calculate the probability of one event given that another related event has already happened.

Example: Conditional Probability in Ball Selection

Suppose we have two boxes: Box 1 with a balls and b white balls, and Box 2 with a black balls and b white balls. We want to calculate the conditional probability of selecting a white ball from Box 1 given that we have moved one ball from Box 1 to Box 2.

Steps to Calculate Conditional Probability in the Ball Selection Problem

Identify Events A and B: Event A: Selecting a white ball from Box 1. Event B: Selecting a ball from Box 1. Calculate ( P(B) ):

There are 2ab total balls in Box 1, so:

[ P(B) frac{2ab}{2a} frac{1}{2} ] Calculate ( P(A cap B) ):

There are a white balls in Box 1, so:

[ P(A cap B) frac{a}{2ab} ] Apply the Formula:

Substituting the values into the formula, we get:

[ P(A|B) frac{P(A cap B)}{P(B)} frac{frac{a}{2ab}}{frac{1}{2}} frac{a}{2ab} times frac{2}{1} frac{a}{2ab} times 2 frac{a}{2a} frac{1}{2} ]

This means the probability of selecting a white ball from Box 1, given that we have selected a ball from Box 1, is ( frac{1}{2} ).

Complex Scenario: Moving a Ball to Another Box

In a more complex scenario, suppose we move one ball from Box 1 to Box 2. The problem now involves calculating the conditional probability of selecting a white ball from Box 1, given that we have moved one ball from Box 1 to Box 2.

Steps to Solve the Complex Problem

Identify Events: Event A: Selecting a white ball from Box 1 after moving a ball. Event B: Selecting a ball from Box 1 and moving it to Box 2. Calculate Probabilities: Event B1: Selecting a white ball from Box 1 and moving it to Box 2. Event B2: Selecting a black ball from Box 1 and moving it to Box 2. Apply the Conditional Probability Formula: ( P(B1) frac{a}{2ab} ) ( P(B2) frac{b}{2ab} ) ( P(A|B1) frac{a-1}{2ab-1} ) ( P(A|B2) frac{b}{2ab-1} ) Combine Probabilities:

The combined probability of selecting a white ball from Box 1 is:

[ P(A|B) P(B1)P(A|B1) P(B2)P(A|B2) ] [ P(A|B) frac{a}{2ab} times frac{a-1}{2ab-1} frac{b}{2ab} times frac{b}{2ab-1} ] [ P(A|B) frac{a(a-1) b^2}{2ab(2ab-1)} ] [ P(A|B) frac{2a(b-a)}{2ab(2ab-1)} ]

Hence, the final answer for the probability of selecting a white ball from Box 1 after moving one ball to Box 2 is:

Final Answer

[ P(A|B) frac{2a(b-a)}{2ab(2ab-1)} ]

This formula provides a detailed approach to solving conditional probability problems involving the transfer of items between different sets.