Problem 31 - 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. \(P(\text { neither is a horse) }\)

Step-by-Step Solution

Answer

The probability that neither tile is a horse is \( \frac{19}{36} \).

Step 1: Calculate Probability No Horse in First Stack

In the first stack, there are 30 tiles with 11 horses. Therefore, the probability of not picking a horse is \( \frac{30 - 11}{30} = \frac{19}{30} \).

Step 2: Calculate Probability No Horse in Second Stack

In the second stack, there are 30 tiles with 5 horses. Thus, the probability of not picking a horse is \( \frac{30 - 5}{30} = \frac{25}{30} = \frac{5}{6} \).

Step 3: Calculate Combined Probability

To find the probability of neither stack showing a horse, multiply the probabilities from Step 1 and Step 2: \( P(\text{neither is a horse}) = \frac{19}{30} \times \frac{5}{6} = \frac{95}{180} \).

Step 4: Simplify Probability

Simplify \( \frac{95}{180} \) by finding the greatest common divisor (GCD) of 95 and 180, which is 5. Divide both the numerator and the denominator by 5 to get \( \frac{19}{36} \).

Key Concepts

Combined Probability

When we talk about combined probability, we're looking at the likelihood of two or more events happening together. In this exercise, we want to find the chance that neither top tile from the stacks is a horse.

To do this, we use the concept of independent probabilities. An independent event means the outcome of one doesn't affect the other. For example:

• The event of choosing a tile from the first stack.
• The event of choosing a tile from the second stack.

To find the overall probability of both events occurring back-to-back, you multiply their separate probabilities. For the first stack, the chance of not getting a horse is \( \frac{19}{30} \). For the second stack, it's \( \frac{25}{30} \). By multiplying these: \[ P(\text{neither is a horse}) = \frac{19}{30} \times \frac{5}{6} = \frac{95}{180} \] Notice, we first simplified \( \frac{25}{30} \) to \( \frac{5}{6} \) before multiplying.

Simplifying Fractions

Simplifying fractions makes them easier to understand and use. It's like reducing a recipe to its essential ingredients.
In our solution, we calculated the combined probability as \( \frac{95}{180} \). This fraction can be hard to interpret, so we simplify it by dividing both the top number (numerator) and the bottom number (denominator) by their greatest common divisor (GCD).

Finding the GCD involves identifying the largest number that divides both numbers evenly. For 95 and 180, their GCD is 5. Thus, we simplify the fraction:

• Divide 95 by 5 to get 19.
• Divide 180 by 5 to get 36.

This gives us \( \frac{19}{36} \). Always remember, a simplified fraction conveys the same value in a clearer, reduced form.

Step-by-Step Solution

Solving probability problems is like following a recipe: step-by-step ensures accuracy and understanding. Here’s how we solved the problem:
1. **Calculate Separate Probabilities**: First, determine the probability of not choosing a horse from each stack. For the first stack, we calculated \( \frac{19}{30} \); for the second, \( \frac{25}{30} = \frac{5}{6} \).
2. **Combine Probabilities**: Multiply the probabilities from each stack to get the combined probability. This results in \( \frac{95}{180} \).
3. **Simplify the Result**: Finally, simplify \( \frac{95}{180} \) by finding the GCD. This gives \( \frac{19}{36} \), a more manageable result.This methodical approach not only helps in clarity but ensures each probability calculation is correct before moving to the next step.