Calculating the Probability of Getting Two Different Colors from a Spinner with Uneven Sections
Working with probability in situations involving balanced and unbalanced scenarios can be a fascinating exercise. This article will explore the probability of getting two different colors (red, blue, and green) when spinning a fair spinner with 11 unequal sections: 4 red, 4 blue, and 3 green. Let's break this problem into steps for clarity and accuracy.
Understanding the Basics
A fair spinner divided into 11 sections with 4 red, 4 blue, and 3 green sections offers a unique challenge in calculating probabilities. Each section is equal in size, but the number of red, blue, and green sections differs. This makes each color outcome have a different probability. Let's dive into the detailed calculation process.
Total Outcomes
When the spinner is spun twice, the total number of outcomes is the product of the number of sections and itself since each spin is independent of the other:
Total Outcomes 11 * 11 121
Favorable Outcomes for Two Different Colors
The favorable outcomes are the cases where the two spins result in different colors. We will consider the possible color pairs and how many outcomes correspond to each pair:
Red and Blue
- First spin: Red (4 ways) and second spin: Blue (4 ways) - First spin: Blue (4 ways) and second spin: Red (4 ways)
Total for Red and Blue 4 * 4 4 * 4 32
Red and Green
- First spin: Red (4 ways) and second spin: Green (3 ways) - First spin: Green (3 ways) and second spin: Red (4 ways)
Total for Red and Green 4 * 3 3 * 4 24
Blue and Green
- First spin: Blue (4 ways) and second spin: Green (3 ways) - First spin: Green (3 ways) and second spin: Blue (4 ways)
Total for Blue and Green 4 * 3 3 * 4 24
Total favorable outcomes for different colors 32 24 24 80
Calculating the Probability
The probability of getting two different colors when spinning the spinner twice is given by the number of favorable outcomes divided by the total number of outcomes:
P2 different colors Number of favorable outcomes / Total outcomes 80 / 121
Therefore, the probability of spinning the spinner twice and getting two different colors is:
pre boxed{80/121}/pre
Optional Alternative Approach:
Another approach involves calculating the probabilities for not getting any two of the same color. Here's how:
Chances of two reds 4/9 * 4/9 16/81 Chances of two blues 2/9 * 2/9 4/81 Chances of two greens 3/9 * 3/9 9/81
Total chances of getting two of the same color 16/81 4/81 9/81 29/81
preChances of not getting two of the same color 81/81 - 29/81 52/81/pre
Thus, the chances of getting two different colors 71/81 or 87.7%.
preP2 different colors 52/81/pre
Both methods provide the same result, confirming the accuracy of our calculations.