Question

A jar contains 10 blue marbles, 5 red marbles, 4 green marbles, and 1 yellow marble. Two marbles are chosen (without replacement). (a) What is the probability that one will be green and the other red? (b) What is the probability that one will be blue and the other yellow?

Short answer

(a) \(\frac{2}{19}\); (b) \(\frac{1}{19}\).

Step-by-step solution

Identify the Total Number of Marbles

First, determine the total number of marbles in the jar. Add together the number of blue, red, green, and yellow marbles: 10 + 5 + 4 + 1 = 20 marbles.

Calculate Total Outcomes for Choosing Two Marbles

Next, calculate the total number of ways to choose 2 marbles from 20. Use the combination formula \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). So, \( \binom{20}{2} = \frac{20!}{2!(20-2)!} = 190 \).

Calculate Probability for One Green and One Red Marble

Calculate the number of ways to choose 1 green and 1 red marble. First, choose 1 green marble from 4: \( \binom{4}{1} = 4 \), and 1 red marble from 5: \( \binom{5}{1} = 5 \). Multiply these results to find the number of favorable outcomes: 4 * 5 = 20. Divide this by the total number of combinations to get the probability: \( \frac{20}{190} = \frac{2}{19} \).

Calculate Probability for One Blue and One Yellow Marble

Calculate the number of ways to choose 1 blue and 1 yellow marble. First, choose 1 blue marble from 10: \( \binom{10}{1} = 10 \), and 1 yellow marble from 1: \( \binom{1}{1} = 1 \). Multiply these results to find the number of favorable outcomes: 10 * 1 = 10. Divide this by the total number of combinations to get the probability: \( \frac{10}{190} = \frac{1}{19} \).

Key concepts

Combinatorics

Combinatorics is a branch of mathematics that deals with counting, arrangement, and combination of objects. In probability theory, combinatorics plays a crucial role as it helps in determining how likely certain events are to occur.

One of the essential tools in combinatorics is the combination formula, denoted as \( \binom{n}{r} \). This formula calculates the number of ways to choose \( r \) objects from \( n \) total objects without regard to order. The formula for this computation is given by:

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

Here, \( n! \) (read as "n factorial") is the product of all positive integers from 1 to \( n \).

In our exercise, when determining the total number of ways to select two marbles from a set of 20, we used the combination formula: \( \binom{20}{2} = 190 \). This number represents all possible ways to pick any two marbles from the jar, irrelevant of their color. The computation serves as the foundation for further probability calculations in this scenario.

Conditional Probability

Conditional probability refers to the likelihood of an event occurring, given that another event has already occurred. In the context of this exercise, if we need to select one green marble and one red marble from a jar, it involves a condition: after one marble is chosen, the probability of the second marble being of the other specified color changes.

However, in our problem, since the probability divides into simple independent binomial events, the steps we use simplify from typical conditional probability methods. While considering multiple events (choosing a green then a red marble or vice versa), we often need to carve out dependencies.

The key is recognizing that when we talk about multiple selections, without replacement, each preceding selection affects the subsequent. Still, computations focus on counting favorable outcomes and dividing them by total possible outcomes.

Probability Formulas

Probability formulas help us measure the likelihood of an event. When dealing with combined independent events in probability, calculations involve finding the number of favorable outcomes over the total number of possible outcomes.

In our exercise, we explore two probabilities:

• One green and one red marble
• One blue and one yellow marble

We start by calculating the favorable outcomes—using combinatorics for each pair of events—then divide by total outcomes to ascertain their probabilities.

For one green and one red, the probability is calculated as:

• Ways to choose one green marble (\( \binom{4}{1} = 4 \))
• Ways to choose one red marble (\( \binom{5}{1} = 5 \))
• Total ways of successfully drawing these: \( 4 \times 5 = 20 \)
• Probability: \( \frac{20}{190} = \frac{2}{19} \)

Similarly, for one blue and one yellow, the probability is:

• Ways to choose one blue marble (\( \binom{10}{1} = 10 \))
• Ways to choose one yellow marble (\( \binom{1}{1} = 1 \))
• Total ways of successfully drawing these: \( 10 \times 1 = 10 \)
• Probability: \( \frac{10}{190} = \frac{1}{19} \)

The probability formula thus allows us to plug in the results of combinatorial calculations, offering a quantitative sense of how likely each scenario is.