Problem 35 - 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. \(P(\text { one is a dog and one is a flower) }\)

Step-by-Step Solution

Answer

The probability is \(\frac{1}{30}\).

Step 1: Identify Possible Outcomes

There are two stacks of tiles, and we need to choose the top tile from each stack. The first stack consists of 30 tiles: 10 dogs, 4 cats, 5 balls, and 11 horses. The second stack also has 30 tiles: 3 flowers, 8 fish, 12 balls, 2 cats, and 5 horses.

Step 2: Determine Favorable Outcomes for a Dog

From the first stack, the favorable outcome is picking a dog. The number of dog tiles is 10. Thus, the probability of picking a dog from the first stack is \(\frac{10}{30} = \frac{1}{3}\).

Step 3: Determine Favorable Outcomes for a Flower

From the second stack, the favorable outcome is picking a flower. The number of flower tiles is 3. Thus, the probability of picking a flower from the second stack is \(\frac{3}{30} = \frac{1}{10}\).

Step 4: Calculate Combined Probability

To find the probability that one tile is a dog and the other is a flower, multiply the probability of each individual event occurring. The probability is \(\frac{1}{3} \times \frac{1}{10} = \frac{1}{30}\).

Key Concepts

Favorable Outcomes

In probability, identifying "favorable outcomes" is the key step that leads us to calculate the likelihood of an event. Favorable outcomes are those outcomes that fulfill the specific conditions of the event we are interested in.
For instance, if you wish to draw a dog from a stack of tiles, all the tiles with a dog's picture on them are considered favorable outcomes for this event.
Here's how it works:

• Identify the total number of outcomes possible. With the tile game, the total number of tiles in each stack is 30.
• Count the specific outcomes that match your event criteria. For choosing a dog, there are 10 dog tiles.
• The probability is the ratio of favorable outcomes to total possible outcomes, which, for a dog, is \( \frac{10}{30} = \frac{1}{3} \).

Understanding favorable outcomes helps us measure how often an event can happen in a given scenario. This is a fundamental step in any probability calculation.

Independent Events

When dealing with probability, it's crucial to determine if events are independent. Independent events mean that the outcome of one event does not affect the outcome of another. In the tile game example, picking a tile from the first stack does not influence what you pick from the second stack, thus they are independent.
Here's why independence matters:

• Independence allows us to calculate probabilities without bearing in mind previous outcomes.
• This simplifies the calculation as events do not overlap; they do not affect each other.
• For example, the probabilities of drawing a dog (\( \frac{1}{3} \)) and a flower (\( \frac{1}{10} \)) remain constant because each stack is decided independently.

Grasping this concept is important because it provides a straightforward path to calculate probabilities in multi-step or multi-part scenarios, such as when combining outcomes.

Combined Probability

To find the "combined probability" of two independent events occurring together, you multiply the probabilities of each event. This approach helps to determine the likelihood of both conditions being met simultaneously, as seen with the game tiles.
The process involves:

• Calculating the probability of each event individually. For a dog, the probability is \( \frac{1}{3} \), and for a flower, it is \( \frac{1}{10} \).
• Since the events are independent, you can multiply these probabilities to find the combined probability. Thus, the probability of drawing one dog and one flower is \( \frac{1}{3} \times \frac{1}{10} = \frac{1}{30} \).
• This calculation tells us how likely it is for both events to occur at the same time when dealing with two independent actions.

Mastering combined probability enables you to understand more complex scenarios where multiple events or conditions might happen concurrently.