Problem 55 - Algebra 2

Question

A gumball machine contains 7 red, 8 orange, 9 purple, 7 white, and 5 yellow gumballs. Tyson buys 3 gumballs. Find each probability, assuming that the machine dispenses the gumballs at random. \(P(3 \text { red })\)

Step-by-Step Solution

Answer

The probability of drawing 3 red gumballs is \(\frac{1}{204}\).

Step 1: Determine Total Number of Gumballs

First, find the total number of gumballs in the machine. Add the numbers of all the colors: \(7 + 8 + 9 + 7 + 5 = 36\). Thus, there are 36 gumballs in total.

Step 2: Calculate the Probability of the First Red Gumball

The probability of getting a red gumball first is the number of red gumballs divided by the total number of gumballs. \(P(1 \text{ red in the first draw}) = \frac{7}{36}\).

Step 3: Calculate the Probability of the Second Red Gumball

After getting 1 red gumball, there are now 6 red gumballs left out of 35 total gumballs. \(P(1 \text{ red in the second draw}) = \frac{6}{35}\).

Step 4: Calculate the Probability of the Third Red Gumball

After drawing the second red gumball, there are 5 red gumballs left out of 34 total gumballs. \(P(1 \text{ red in the third draw}) = \frac{5}{34}\).

Step 5: Calculate the Overall Probability of Drawing 3 Red Gumballs

Multiply the probabilities obtained in Steps 2, 3, and 4 to find the overall probability of drawing 3 red gumballs: \[P(3 \text{ red }) = \frac{7}{36} \times \frac{6}{35} \times \frac{5}{34} = \frac{210}{42840}\].This fraction can be simplified to \(\frac{1}{204}\).

Key Concepts

Combinations

Combinations are a fundamental concept in probability and combinatorics, helping us determine how many ways we can select items from a larger set. Unlike permutations, combinations do not take the order of selection into account. When calculating how many ways we can choose a set number of items, we use the combination formula, which is represented as:

• \(C(n, r) = \frac{n!}{r!(n-r)!}\)

Here, \(n\) is the total number of items to choose from, and \(r\) is the number of items to choose.
Factorials, denoted by \(!\), play an important role in this calculation, representing the product of all positive integers up to a given number.
This concept is frequently used when, for instance, we want to choose a subset of gumballs from the gumball machine without considering the order of selection. In our specific problem, however, the focus is on permutations rather than combinations, as the order of selection impacts the outcome.

Permutations

Permutations are all about arrangements where the order does matter. When dealing with gumballs, if we care about the order in which colors are dispensed, permutations are applicable. The formula for permutations is used when we need to arrange \(r\) items from a set of \(n\) total items in an ordered manner:

• \(P(n, r) = \frac{n!}{(n-r)!}\)

In this scenario with the gumball machine, although combinations of colors matter more, sometimes permutations might be needed in different contexts. For instance, some games might require considering the order in which you get specific-colored gumballs. While solving the problem of finding 3 red gumballs, the sequence matters when calculating probabilities since each draw affects the following.

Gumball Machine

A gumball machine is a delightful example of understanding probability in a tangible form. It typically holds a variety of colored gumballs, making it an excellent tool to introduce complex concepts such as independent and dependent events.
In this exercise, the gumball machine serves as a model for calculating probabilities when drawing items randomly: the machine dispenses gumballs without replacement, altering the total count with each draw.
This changing total introduces dependencies between draws rather than independent events, affecting our calculations of probabilities for particular outcomes, such as drawing a specific color multiple times.

Independent Events

In probability, independent events are those whose outcomes do not affect one another. If one event occurs, it doesn't influence another. For instance, tossing a coin twice results in independent events, as the result of the first toss doesn't impact the second.
However, with the gumball machine, each draw reduces the total number of gumballs and potentially the number of a specific color. In this context, drawing gumballs represents dependent events, as each draw affects subsequent probabilities. Understanding the difference is crucial.
In our situation, each time you get a red gumball, the likelihood of the next event (drawing another red) changes, since the count and total available gumballs are reduced, showcasing dependency.