Problem 43 - Calculus for Biology and Medicine

Question

An urn contains four green, six blue, and two red balls. You take three balls out of the urn without replacement. What is the probability that all three balls are of different colors?

Step-by-Step Solution

Answer

The probability is \(\frac{12}{55}\).

Step 1: Determine Total Possibilities

First, we need to calculate the total number of ways to draw three balls from the urn. The urn has a total of \(4 + 6 + 2 = 12\) balls. The number of ways to choose 3 balls out of 12 is given by the binomial coefficient \(\binom{12}{3}\). Thus, \(\binom{12}{3} = \frac{12 \cdot 11 \cdot 10}{3 \cdot 2 \cdot 1} = 220\).

Step 2: Calculate Desired Outcomes

Now, find the number of ways to choose one ball of each color. We can choose 1 green, 1 blue, and 1 red ball. The number of ways to choose 1 green ball from 4 is \(\binom{4}{1} = 4\). For blue, it's \(\binom{6}{1} = 6\). For red, it's \(\binom{2}{1} = 2\). Multiply these together to find the total number of ways: \(4 \times 6 \times 2 = 48\).

Step 3: Calculate Probability

Now that we have the number of favorable outcomes (48) and total outcomes (220), the probability that all three balls are of different colors is the ratio of these two numbers: \(\frac{48}{220}\). Simplify the fraction to get \(\frac{24}{110}\), and further simplify to \(\frac{12}{55}\).

Step 4: Confirm Simplification

Ensure that \(\frac{12}{55}\) is in its simplest form. The greatest common divisor of 12 and 55 is 1, so the fraction is already simplified.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that focuses on counting, arranging, and finding patterns among sets. In our exercise, we use combinatorics to determine the number of ways to draw and arrange balls of different colors from the urn. Since we are taking balls without replacement, each selection affects the next choice, thus complicating simple counting methods. To solve this kind of problem, we often rely on formulas like the binomial coefficient, which helps us calculate combinations efficiently. Without combinatorics, it would be nearly impossible to handle such problems manually, especially as the number of items (in this case, balls) increases. This is because the direct counting method can quickly become overwhelming, making combinatorial techniques invaluable in simplifying the process.

Binomial coefficient

The binomial coefficient is critical for solving problems related to selecting items from a larger group, such as selecting balls from the urn in our problem. It is represented as \( \binom{n}{k} \), where \( n \) is the total number of items, and \( k \) is the number of items to choose. The formula for the binomial coefficient is:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

In our exercise, \( \binom{12}{3} \) calculates the number of ways to choose 3 balls from a total of 12 balls. We simplify this by understanding that 12 factorial (\( 12! \)) is the product of descending natural numbers from 12, divided by both 3 factorial (\( 3! \)) and the rest (\( 9! \) not chosen), which is mathematics of arranging groups without regard to specific order. The importance of understanding binomial coefficients is rooted in their ability to simplify complex arrangements and selections, which are prevalent in many probability and statistical problems.

Probability of independent events

Probability of independent events is foundational when tackling problems that involve random selection or arrangement. An independent event means that the occurrence or outcome of one event does not affect the other. In our exercise, although we're dealing with probability, the events aren't independent since drawing one ball affects the probability of future draws. Nonetheless, calculating probabilities overall relies heavily on understanding when events influence each other and when they do not.For the probability that all three balls are of different colors, we calculated through dependent events because colors selected in each draw depend on previous draws. Here, we use the total probability formula:

• Probability = \( \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \)

Despite the dependency in each draw, this formula gives us a generalized approach for estimating the chance of a particular arrangement. Grasping this reasoning can provide insights into more complex problems involving multiple steps or conditions.