The Probability of Drawing Pennies from Brad's Pocket: A Common Homework Problem
Brad has 4 nickels and 2 pennies in his pocket. This sets the stage for an interesting probability question: If Brad takes out one coin and then replaces it before taking out a second coin, what is the probability that both coins are pennies?
At first glance, the problem might seem straightforward, but understanding the nuances of dependent versus independent events is crucial.
Understanding the Basics
Let's break down the situation. Brad has 6 coins in total: 4 nickels and 2 pennies. When he takes out a coin and then replaces it, each draw is independent of the previous one. This is important because the probability of drawing a penny remains the same for each draw.
Independent Draw Analysis
First, let's calculate the probability of drawing a penny in the first draw. Since there are 2 pennies out of 6 coins, the probability is:
(frac{2}{6} frac{1}{3})
After replacing the coin, the total number of coins remains 6, and the probability of drawing a penny again is still (frac{1}{3}).
Therefore, the probability of drawing two pennies in a row (considering each draw is independent) is:
(frac{1}{3} times frac{1}{3} frac{1}{9})
Why This Problem Is Often a Homework Assignment
The problem is a common homework question because it tests a student's understanding of independent events and probability calculations. It's a straightforward problem with a clear solution, making it perfect for educational purposes and for assessing students’ grasp of basic probability concepts.
Hint to Students
One hint that can guide students is to recognize whether the events are independent. In this case, because the coin is replaced after the first draw, the events are independent. This is different from scenarios where the coin is not replaced (dependent events), and the probability changes after the first draw.
Encouraging students to ask themselves whether the trials are dependent or independent can help them solve similar problems in the future. Such self-reflection is a valuable skill in probability and statistics.
Conclusion
The probability of drawing two pennies in a row from Brad's pocket, with replacement, is (frac{1}{9}). Mastery of such problems is crucial for building a strong foundation in probability. It's always important for students to understand the concepts behind the solutions and not just memorize the answers. By doing the work and learning the material, students can effectively tackle more complex probability questions in the future.