Problem 30 - Algebra 2

Question

GAMES For Exercises \(30-35\) , use the following information. A certain game has two stacks of 30 tiles with pictures on them. In the first stack of tiles, there are 10 dogs, 4 cats, 5 balls, and 11 horses. In the second stack of tiles, there are 3 flowers, 8 fish, 12 balls, 2 cats, and 5 horses. The top tile in each stack is chosen. Find each probability. \(P(\text { each is a ball })\)

Step-by-Step Solution

Answer

The probability that each tile is a ball is \( \frac{1}{15} \).

Step 1: Determine Total Number of Tiles in Each Stack

The problem states that there are two stacks of 30 tiles each. Therefore, the total number of tiles in each stack is 30.

Step 2: Count Number of Ball Tiles in Each Stack

In the first stack, there are 5 ball tiles. In the second stack, there are 12 ball tiles.

Step 3: Calculate Probability of Drawing a Ball from Each Stack Separately

The probability of drawing a ball from the first stack is \( P_1 = \frac{5}{30} = \frac{1}{6} \). The probability of drawing a ball from the second stack is \( P_2 = \frac{12}{30} = \frac{2}{5} \).

Step 4: Calculate Combined Probability of Each Tile Being a Ball

The probability that both tiles drawn are balls is the product of their individual probabilities: \( P(\text{both balls}) = P_1 \times P_2 = \frac{1}{6} \times \frac{2}{5} = \frac{1}{15} \).

Key Concepts

Independent Events

In probability, understanding independent events is crucial. Independent events are those where the outcome of one event does not affect the outcome of another. For instance, if you flip a coin and roll a die, the result of the coin flip has no impact on the result of the die roll. Each event happens without any influence from the other.
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In the context of the given exercise, each draw from the two separate stacks of tiles is an independent event. Drawing a ball from the first stack does not affect the probability of drawing a ball from the second stack. Each tile is chosen randomly and independently from its respective stack, making these independent events a key part of the calculation.

Multiplication Rule

The multiplication rule is a fundamental concept in probability useful for finding the probability of two independent events both occurring. According to the rule, if the probability of event A occurring is \(P(A)\) and event B occurring is \(P(B)\), then the probability of both events A and B occurring is given by the formula:

• \(P(A \text{ and } B) = P(A) \times P(B)\)

This rule applies specifically when events A and B are independent.
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In the exercise, we use the multiplication rule to determine the probability of drawing a ball tile from both stacks. We calculate the probability for each stack separately and multiply them together:

• \(P_1 = \frac{1}{6}\) from the first stack
• \(P_2 = \frac{2}{5}\) from the second stack

Thus, the probability that both drawn tiles are balls is \(\frac{1}{6} \times \frac{2}{5} = \frac{1}{15}\).

Sample Space

The sample space is the set of all possible outcomes of an experiment or a situation. In probability, defining the sample space is essential to understanding and calculating probabilities accurately.
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For our exercise, the sample space for each stack is the full assortment of tiles, totaling 30 per stack. This means there are potential outcomes associated with each of the 30 tiles in both stacks. When considering the event where a ball is drawn, the specific outcomes we are interested in are the selectable ball tiles within each stack,

• 5 ball tiles in the first stack
• 12 ball tiles in the second stack

The sample space helps clarify the probability calculation as it establishes the total number of events that could happen when a tile is drawn from a stack.

Combinatorics

Combinatorics involves the counting, arrangement, and combination of objects. In probability, it helps us determine the number of possible outcomes and events, usually when forming combinations or sequences.
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In scenarios like drawing tiles from a stack, basic combinatorial principles are used implicitly. While the exercise at hand doesn’t require complex combinatorial calculations, understanding that there are different ways to choose and combine tiles is rooted in combinatorics. Here, we focus on how many ways we can draw a particular type of tile from each stack. For more complex problems, combinatorics can be a deep and powerful tool to figure out probabilities of more complicated events involving larger selections or arrangements of objects.