Probability Of Selecting 3 Orange And 2 Yellow Balls From An Urn
In probability theory, we often encounter problems involving selecting items from a set, and calculating the probability of specific outcomes. These problems often involve combinations, as the order of selection typically doesn't matter. Let's delve into a classic probability problem involving an urn filled with balls of different colors.
This article aims to provide a comprehensive understanding of how to calculate probabilities in scenarios involving combinations, specifically focusing on the problem of selecting balls from an urn. We will break down the problem step-by-step, explaining the underlying concepts and formulas involved. Our primary focus will be on a scenario where an urn contains 7 orange balls and 5 yellow balls, and we want to determine the probability of selecting 3 orange balls and 2 yellow balls when 5 balls are chosen at random. We will explore the concepts of combinations, total possible outcomes, and favorable outcomes, ultimately arriving at the solution and rounding it to three decimal places. This exploration will not only help in solving this specific problem but also provide a framework for tackling similar probability questions.
Consider an urn that contains 7 orange balls and 5 yellow balls. Suppose Vince chooses 5 balls at random from this urn. What is the probability that he will select exactly 3 orange balls and 2 yellow balls? We will round the final answer to three decimal places.
Before we dive into the solution, it's crucial to understand the concept of combinations. In combinatorics, a combination is a selection of items from a set where the order of selection does not matter. The number of ways to choose k items from a set of n items is denoted as "n choose k" or C(n, k), and it is calculated using the following formula:
C(n, k) = n! / (k! * (n - k)!)
where "!" denotes the factorial function. The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, 5! = 5 * 4 * 3 * 2 * 1 = 120.
Combinations are fundamental in probability calculations when the order of selection is irrelevant. Understanding this concept is key to tackling problems like the one presented, where we are concerned with the number of ways to select a specific number of balls of each color, regardless of the order in which they are chosen. The formula for combinations allows us to accurately calculate the number of possible selections, which is a crucial step in determining the probability of a particular outcome. By grasping the principles of combinations, we can confidently approach and solve a wide range of probability problems.
To find the probability, we first need to determine the total number of ways Vince can choose 5 balls from the urn. The urn contains a total of 7 orange balls + 5 yellow balls = 12 balls. Vince is choosing 5 balls out of these 12 balls. The order in which he chooses the balls does not matter, so this is a combination problem.
We need to calculate C(12, 5), which represents the number of ways to choose 5 balls from a set of 12 balls. Using the combination formula:
C(12, 5) = 12! / (5! * (12 - 5)!) = 12! / (5! * 7!)
Now, let's compute the factorials:
12! = 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
5! = 5 * 4 * 3 * 2 * 1 = 120
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040
Substitute these values into the combination formula:
C(12, 5) = 12! / (5! * 7!) = (12 * 11 * 10 * 9 * 8 * 7!) / (120 * 7!) = (12 * 11 * 10 * 9 * 8) / 120
Simplify the expression:
C(12, 5) = (12 * 11 * 10 * 9 * 8) / (5 * 4 * 3 * 2 * 1) = 95040 / 120 = 792
Therefore, there are 792 total possible ways to choose 5 balls from the urn. This number represents the denominator in our probability calculation, as it encompasses all possible outcomes of the ball selection process. Understanding how to calculate the total possible outcomes is essential for determining the probability of specific events, such as selecting a particular combination of orange and yellow balls. This step sets the foundation for calculating the favorable outcomes and, ultimately, the desired probability.
Next, we need to calculate the number of ways Vince can select exactly 3 orange balls and 2 yellow balls. This involves calculating the combinations for each color separately and then multiplying them together.
First, let's find the number of ways to choose 3 orange balls from the 7 orange balls in the urn. This is C(7, 3):
C(7, 3) = 7! / (3! * (7 - 3)!) = 7! / (3! * 4!)
Compute the factorials:
7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040
3! = 3 * 2 * 1 = 6
4! = 4 * 3 * 2 * 1 = 24
Substitute these values into the combination formula:
C(7, 3) = 7! / (3! * 4!) = 5040 / (6 * 24) = 5040 / 144 = 35
So, there are 35 ways to choose 3 orange balls from the 7 available.
Now, let's find the number of ways to choose 2 yellow balls from the 5 yellow balls in the urn. This is C(5, 2):
C(5, 2) = 5! / (2! * (5 - 2)!) = 5! / (2! * 3!)
Compute the factorials:
5! = 5 * 4 * 3 * 2 * 1 = 120
2! = 2 * 1 = 2
3! = 3 * 2 * 1 = 6
Substitute these values into the combination formula:
C(5, 2) = 5! / (2! * 3!) = 120 / (2 * 6) = 120 / 12 = 10
Thus, there are 10 ways to choose 2 yellow balls from the 5 available.
To find the total number of favorable outcomes (selecting 3 orange balls and 2 yellow balls), we multiply the number of ways to choose orange balls by the number of ways to choose yellow balls:
Favorable outcomes = C(7, 3) * C(5, 2) = 35 * 10 = 350
Therefore, there are 350 favorable outcomes for selecting 3 orange balls and 2 yellow balls. This number represents the numerator in our probability calculation, as it reflects the specific outcome we are interested in. Calculating favorable outcomes involves understanding how to combine different selections to achieve the desired result, which is a crucial skill in probability problems involving multiple conditions or categories.
Now that we have calculated the total possible outcomes and the number of favorable outcomes, we can calculate the probability of Vince selecting 3 orange balls and 2 yellow balls. The probability is the ratio of favorable outcomes to total possible outcomes:
Probability = (Favorable Outcomes) / (Total Possible Outcomes)
Probability = 350 / 792
Now, let's calculate the value and round it to three decimal places:
Probability ≈ 0.4419191919...
Rounded to three decimal places, the probability is approximately 0.442.
Thus, the probability that Vince will select 3 orange balls and 2 yellow balls is approximately 0.442. This result represents the likelihood of the specific event occurring out of all possible outcomes. Calculating the probability by dividing favorable outcomes by total possible outcomes is the fundamental principle in probability theory, and this step provides the final answer to our problem. By understanding how to calculate probabilities in this context, we can apply the same principles to a variety of similar scenarios.
In conclusion, the probability that Vince will select 3 orange balls and 2 yellow balls from the urn is approximately 0.442. This result was obtained by systematically calculating the total possible outcomes and the favorable outcomes using the principles of combinations. The combination formula, C(n, k) = n! / (k! * (n - k)!), played a crucial role in determining the number of ways to select balls without regard to order.
We first calculated the total possible ways to choose 5 balls from the 12 balls in the urn, which was C(12, 5) = 792. Then, we calculated the number of ways to choose 3 orange balls from 7 and 2 yellow balls from 5, which were C(7, 3) = 35 and C(5, 2) = 10, respectively. Multiplying these values gave us the number of favorable outcomes, 35 * 10 = 350.
Finally, we divided the number of favorable outcomes by the total possible outcomes to find the probability: 350 / 792 ≈ 0.442. This problem demonstrates the application of combinatorics in probability and highlights the importance of understanding combinations in solving such problems.
The process of solving this problem provides a clear framework for tackling similar probability questions involving combinations. By breaking down the problem into smaller steps—calculating total possible outcomes, determining favorable outcomes, and then finding the ratio—we can approach complex scenarios with confidence. The ability to apply these principles is invaluable in various fields, from statistics and data analysis to everyday decision-making. This exercise not only provides a specific solution but also enhances our understanding of probabilistic reasoning and problem-solving skills.
By mastering these techniques, individuals can better analyze and predict outcomes in situations involving uncertainty, making informed choices based on sound probability calculations. The concepts discussed in this article serve as a foundation for further exploration of probability theory and its diverse applications.
