Question

A bag contains 5 white balls, and 6 green balls. Two balls are drawn from the bag one after another, what is the probability that both the balls are white? (the first ball is replaced before drawing the second ball) (1) \(2 / 11\) (2) \(1 / 11\) (3) \(\frac{3}{121}\) (4) \(\frac{25}{121}\)

Short answer

Answer: \(\frac{25}{121}\)

Step-by-step solution

Identify the number of outcomes for each draw

There are 5 white balls and 6 green balls in the bag. So, the number of outcomes for each draw is 11 since the first ball is replaced before drawing the second ball.

Calculate the probability of drawing a white ball

The probability of drawing a white ball is the number of white balls (5) divided by the total number of balls (11). So, the probability of drawing a white ball is \(\frac{5}{11}\).

Calculate the probability of drawing two white balls

Since the two events are independent (i.e., the first ball is replaced before drawing the second ball), we can simply multiply the probabilities of each event to get the joint probability. Therefore, the probability of drawing two white balls is \(\frac{5}{11}\times\frac{5}{11}=\frac{25}{121}\). So, the correct answer is (4) \(\frac{25}{121}\).

Key concepts

Independent Events

In probability theory, an event is said to be independent if the occurrence of one event does not affect the occurrence of another. This is an important concept because it allows us to simplify the calculation of the probability of multiple events happening. In the problem we are considering, the events are drawing balls from a bag. Each time a ball is drawn and replaced, it becomes an independent event. This means the previous draw does not influence the outcome of the subsequent draw. When calculating probabilities for independent events, we can multiply the probabilities of individual events to find the probability of their joint occurrence.

• Example: In our exercise, drawing two white balls is independent. So, the probability is the product of each draw's probability.
• This is mathematically expressed as: \( P(A \ \text{and} \ B) = P(A) \cdot P(B) \)

Replacement in Probability

Replacement in probability means that after an item is drawn from a collection, it is put back before the next item is drawn. This affects the probability calculations because it keeps the total number of items constant for each draw. In our example, a ball is drawn from the bag and then replaced. This ensures that the probability remains the same for each draw since the number of balls in the bag doesn't change. Without replacement, the conditions would change, leading to dependent events. Using replacement ensures the independence of each event, allowing us to easily calculate the probabilities by multiplying the individual probabilities without needing complex calculations.

Probability of Drawing Balls

Calculating the probability of drawing certain balls from a set involves understanding the number of favorable outcomes over the total number of possible outcomes. In our scenario, we have two kinds of balls: white and green.

• Total number of balls = 11 (5 white, 6 green)
• Probability of drawing a white ball = number of white balls / total balls \(= \frac{5}{11}\)

To find the probability of drawing two white balls in succession with replacement, multiply the probability of drawing a white ball twice, \(\frac{5}{11} \times \frac{5}{11} = \frac{25}{121}\). This calculation demonstrates how replacement ensures that probability calculations remain consistent for each event.

Mathematics Class Level Problems

Mathematics problems involving probability often help students understand the foundational concepts of random events and their outcomes. These problems are usually designed with a focus on comprehensible numbers and straightforward processes. These types of problems, such as our example involving drawing balls, teach students how to use simple probability rules, multiplication for independent events, and the concept of replacement. They build critical thinking and problem-solving skills by showing how different mathematical principles come into play in real-world scenarios.

• Core skills developed: computational accuracy, understanding independent events, and manipulating fractions.
• Objective: Enable students to grasp not just the process but also the logic behind probability calculations.

By consistently working through these types of problems, students can improve their mathematical proficiency and prepare for more advanced topics.