Question

A candy dish contains five brown and three red M\&Ms. A child selects three M&Ms without checking the colors. Use this information to answer the questions in Exercises \(18-21 .\) What is the probability that there are two brown and one red M&Ms in the selection?

Short answer

Answer: The probability is 15/28.

Step-by-step solution

Find the total number of M&Ms and total ways to select 3 M&Ms.

In the dish, she has \(5\) brown and \(3\) red M&Ms, which sums up to a total of \(5 + 3 = 8\) M&Ms. To find the number of ways to select \(3\) M&Ms from the \(8\) M&Ms, we can use the combination formula: \(C(n, k) = \frac{n!}{k!(n-k)!}\), where \(n\) is the total number of M&Ms and \(k\) is the number of M&Ms she picks. So the total number of ways to select \(3\) M&Ms is: \(C(8, 3) = \frac{8!}{3!(8-3)!} = {\binom{8}{3}} = 56\).

Find the number of favorable outcomes (2 brown and 1 red M&Ms).

To find the number of ways to select \(2\) brown M&Ms from the \(5\) brown M&Ms, we use the combination formula again: \(C(5, 2) = \frac{5!}{2!(5-2)!} = {\binom{5}{2}} = 10\). Similarly, to find the number of ways to select \(1\) red M&M from the \(3\) red M&Ms, we use the combination formula: \(C(3, 1) = \frac{3!}{1!(3 - 1)!} = {\binom{3}{1}} = 3\). Now, we need to find the ways to pick \(2\) brown and \(1\) red M&Ms. Since these are independent events, we can multiply the number of ways to pick \(2\) brown and \(1\) red M&Ms: \(10 \times 3 = 30\).

Calculate the probability of getting 2 brown and 1 red M&Ms.

Now that we know the total number of ways to choose 3 M&Ms and the number of ways to choose 2 brown and 1 red M&Ms, we can find the probability by dividing the favorable outcomes by the total outcomes: \(P(\text{2 brown, 1 red}) = \frac{\text{number of favorable outcomes}}{\text{number of total outcomes}} = \frac{30}{56} = \frac{15}{28}\). So, the probability of picking \(2\) brown and \(1\) red M&Ms is \(\frac{15}{28}\).

Key concepts

Combinatorics

Understanding combinatorics is essential when it comes to solving probability problems involving the selection of items, such as the given candy dish problem. Combinatorics is a field of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It provides tools for counting the numerous configurations of a collection of objects that satisfy specific criteria.

For instance, when considering the selection of M&Ms from the candy dish, we're dealing with a combinatorial problem. We want to know how many different ways we can choose three candies from a set of eight. One of the key tools in combinatorics is the use of combinations, which helps us calculate how many different groups of items can be formed from a larger set when the order of the items doesn’t matter. The formula to calculate combinations, as seen with the M&M exercise, is expressed using factorial notation, which will be explained in the next section.

Factorial Notation

Factorial notation is a mathematical concept used heavily in probability and combinatorics, serving as the cornerstone for calculating permutations and combinations. The notation for a factorial is an exclamation point \batchmode ! \break , and it represents the product of all positive integers up to a given number. Formally, for a nonnegative integer \(n\), the factorial of \(n\) (denoted as \(n!\)) is defined as:
\batchmode \break \[n! = n \times (n - 1) \times (n - 2) \times \dots \times 2 \times 1\].
By convention, \(0! = 1\), which facilitates formulations in combinatorial mathematics.

Understanding factorials is crucial because they are part of the formula used to calculate combinations. For example, in our exercise, the combination formula is dependent on factorial calculations. This is evident when we calculate the total number of ways to select three M&Ms from eight, and further, when we find the number of ways to select two brown M&Ms from five.

Why is Factorial Important in Probability?

In probability, factorial notation allows us to efficiently determine the number of possible arrangements and selections in a set, which is fundamental when the solution hinges on understanding all possible outcomes.

Probability Calculation

Probability calculation is the process used to determine the likelihood of a particular event happening. The basic formula for probability is:\[P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\]
In the context of the M&Ms exercise, we apply this concept to find the probability of selecting two brown and one red M&M. The calculation involves two parts: firstly determining the total number of outcomes, which is based on the combination of selecting three M&Ms out of eight without regard to order.

Secondly, we identify the favorable outcomes, the combinations of selecting exactly two brown and one red M&M. Multiplying the number of ways to pick two brown M&Ms with the number of ways to pick one red M&M gives us the total favorable outcomes. Finally, the probability is determined by dividing the number of favorable outcomes by the total number of outcomes. This exercise beautifully demonstrates the practical application of probability calculation in combinatorial problems.

Improving Probability Understanding

To enhance understanding of probability, it's helpful to visualize the problem, such as by drawing a diagram or picturing the M&M selection process. Additionally, hands-on activities, such as physically withdrawing candies from a bowl, can solidify comprehension of combinatorial probabilities for many learners.