Probability Of Drawing A Blue And Red Ball

by Rajiv Sharma 43 views

Let's dive into a probability problem involving drawing balls from two bags. This is a classic scenario in probability theory, and understanding how to solve it can be super useful for various real-world applications. In this article, we'll break down the problem step by step, making sure we understand every aspect before arriving at the final solution. So, let's get started!

Understanding Probability

Before we jump into the problem, let’s quickly recap what probability actually means. Simply put, probability is the measure of the likelihood that an event will occur. It's quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. For example, a probability of 0.5 (or 50%) means there’s an equal chance of the event happening or not happening.

The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. Mathematically, we represent this as:

P(Event)=NumberextofextFavorableextOutcomesTotalextNumberextofextPossibleextOutcomesP(Event) = \frac{Number ext{ } of ext{ } Favorable ext{ } Outcomes}{Total ext{ } Number ext{ } of ext{ } Possible ext{ } Outcomes}

Independent Events

Now, a key concept we'll use in our problem is that of independent events. Two events are said to be independent if the occurrence of one doesn’t affect the occurrence of the other. For instance, if you flip a coin and then roll a die, the outcome of the coin flip doesn’t influence the outcome of the die roll. These are independent events.

When we want to find the probability of two independent events both happening, we multiply their individual probabilities. If event A has a probability $P(A)$ and event B has a probability $P(B)$, then the probability of both A and B occurring is:

P(AextandextB)=P(A)×P(B)P(A ext{ } and ext{ } B) = P(A) \times P(B)

This is a fundamental rule in probability and is crucial for solving problems like the one we're about to tackle.

The Ball-Drawing Problem: Setting the Stage

Let's consider our specific problem. We have two bags, Bag A and Bag B, each containing balls of different colors. We're given the following information:

  • The probability of drawing a blue ball from Bag A is $1/5$.
  • The probability of drawing a red ball from Bag B is $2/3$.

The question we need to answer is: What is the probability of drawing a blue ball from Bag A and a red ball from Bag B?

This problem involves two separate events: drawing a blue ball from Bag A and drawing a red ball from Bag B. We need to determine if these events are independent and, if so, apply the rule for calculating the probability of independent events occurring together.

Are the Events Independent?

The first thing we need to figure out is whether the events of drawing a ball from Bag A and drawing a ball from Bag B are independent. Think about it this way: does the color of the ball you draw from Bag A have any effect on the color of the ball you draw from Bag B? The answer is no. The two bags are separate, and the draws are unrelated.

Since the outcome of drawing a ball from Bag A doesn't influence the outcome of drawing a ball from Bag B, these events are indeed independent. This means we can use the rule for multiplying probabilities of independent events to find the combined probability.

Solving the Problem

Okay, now that we've established the events are independent, let's get down to solving the problem. We're given:

  • Probability of drawing a blue ball from Bag A: $P(Blue ext{ } from ext{ } A) = \frac{1}{5}$
  • Probability of drawing a red ball from Bag B: $P(Red ext{ } from ext{ } B) = \frac{2}{3}$

We want to find the probability of both events happening: drawing a blue ball from Bag A and drawing a red ball from Bag B. Using the rule for independent events, we multiply the individual probabilities:

P(BlueextfromextAextandextRedextfromextB)=P(BlueextfromextA)×P(RedextfromextB)P(Blue ext{ } from ext{ } A ext{ } and ext{ } Red ext{ } from ext{ } B) = P(Blue ext{ } from ext{ } A) \times P(Red ext{ } from ext{ } B)

Now, plug in the given probabilities:

P(BlueextfromextAextandextRedextfromextB)=15×23P(Blue ext{ } from ext{ } A ext{ } and ext{ } Red ext{ } from ext{ } B) = \frac{1}{5} \times \frac{2}{3}

Multiply the fractions:

P(BlueextfromextAextandextRedextfromextB)=1×25×3=215P(Blue ext{ } from ext{ } A ext{ } and ext{ } Red ext{ } from ext{ } B) = \frac{1 \times 2}{5 \times 3} = \frac{2}{15}

So, the probability of drawing a blue ball from Bag A and a red ball from Bag B is $\frac{2}{15}$.

Common Mistakes to Avoid

When dealing with probability problems, it's easy to make small errors that can lead to the wrong answer. Here are a few common mistakes to watch out for:

  1. Forgetting to Check for Independence: Always make sure the events are truly independent before multiplying probabilities. If events are dependent (the outcome of one affects the other), you'll need to use a different approach.

  2. Adding Probabilities Incorrectly: Probabilities are added when you want to find the probability of either one event or another happening (and they are mutually exclusive). When you want to find the probability of both events happening, you typically multiply (if they're independent).

  3. Misunderstanding the Question: Read the problem carefully! Make sure you understand exactly what probability you're being asked to find. In our case, it was the probability of both events occurring, not just one or the other.

  4. Arithmetic Errors: Simple mistakes in multiplication or division can throw off your answer. Double-check your calculations to make sure everything is correct.

By keeping these common pitfalls in mind, you'll be well-equipped to tackle probability problems with confidence!

Real-World Applications of Probability

You might be wondering,