Problem 56 - 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. \(P(2 \text { white, } 1 \text { purple })\)

Step-by-Step Solution

Answer

The probability is \( \frac{9}{340} \).

Step 1: Determine Total Gumballs

First, calculate the total number of gumballs in the machine. Add all the gumballs together: 7 (red) + 8 (orange) + 9 (purple) + 7 (white) + 5 (yellow) = 36 gumballs.

Step 2: Calculate Total Ways to Choose 3 Gumballs

Find the total number of ways to choose 3 gumballs from 36. This can be calculated using combination formula \( \binom{n}{k} \), where \( n \) is the total number of gumballs and \( k \) is the number of gumballs to be selected: \( \binom{36}{3} = \frac{36 \times 35 \times 34}{3 \times 2 \times 1} = 7140 \).

Step 3: Calculate Ways to Select 2 White Gumballs

Calculate the number of ways to select 2 white gumballs out of 7. Using the combination formula: \( \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21 \).

Step 4: Calculate Ways to Select 1 Purple Gumball

Calculate the number of ways to select 1 purple gumball out of 9. Using the combination formula: \( \binom{9}{1} = 9 \).

Step 5: Calculate Specific Combination Probability

Calculate the probability of selecting 2 white and 1 purple gumball together by multiplying the combinations for white and purple, then dividing by the total combinations: \( \frac{21 \times 9}{7140} = \frac{189}{7140} \).

Step 6: Simplify Probability Fraction

Simplify the fraction \( \frac{189}{7140} \). The greatest common divisor of 189 and 7140 is 21. Divide both the numerator and the denominator by 21: \( \frac{9}{340} \).

Key Concepts

Combinatorics

Combinatorics is a fascinating branch of mathematics that deals with counting and arranging objects. It's like the art of finding patterns and making sense out of how things can be grouped together. In the context of probability and statistics, combinatorics helps us understand how likely certain outcomes are in scenarios involving multiple elements.

In everyday terms, combinatorics might be like figuring out how many different outfits you can make from a selection of shirts and pants, or how many ways you can arrange books on a shelf. Moreover, whether you are dealing with a colorful array of gumballs or a deck of cards, combinatorics allows you to calculate the number of possible configurations or selections from these choices.

• It can involve permutations, which is about arranging objects in sequence.
• Meanwhile, combinations are about selecting objects without worrying about the order.

For gumballs, where the order of picking doesn't matter, combinations are key. Combinatorics provides the mathematical foundation to determine how many ways you can select a specific group from a larger set.

Combination Formula

The combination formula is a crucial concept in combinatorics. It provides a way to calculate the number of ways to select a group of items from a larger set, where the order of selection does not matter. The formula is expressed as:\[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]Here, \( n \) stands for the total number of items to choose from, and \( k \) is the number of items to select. The exclamation mark denotes a factorial, which means multiplying a series of descending natural numbers.

Imagine you have a large bowl of different colored gumballs. If you want to know how many ways you can randomly select 3 gumballs from a total of 36, the combination formula becomes incredibly helpful in resolving this. It allows you to calculate this without worrying about the sequence in which you pick the gumballs.

• For example, selecting 3 gumballs would look like this: \( \binom{36}{3} \).
• This can be expanded as \( \frac{36 \times 35 \times 34}{3 \times 2 \times 1} = 7140 \).

Understanding this formula is pivotal for solving problems that involve selecting objects from a collection, especially when you're dealing with random selections.

Random Selection

Random selection is a concept where each item in a group has an equal chance of being chosen. This principle is often applied in situations to ensure fairness and avoid bias, such as a gumball machine dispensing a random mix of gumballs.

When you buy gumballs from a machine, you have no control over which colors will pop out. This randomness is what makes using probability calculations and combinatorics appropriate for figuring out the likelihood of getting certain combinations of gumballs. By understanding random selection, you can predict outcomes based on the total number of possibilities.

• In the example of selecting 2 white and 1 purple gumball from a machine, you're combining multiple random selections.
• Your task is to count how many successful scenarios match the desired outcome compared to the total number of possible outcomes.

Using combinations helps you in calculating how different choices could potentially occur in random scenarios, providing a deeper insight into probability and chance.