A Jar Consists Of Blue, Red And Green Marbles. The Probability Of Each Of Them Is 2/7 , 3/k And 5/2k.
Understanding probabilities is fundamental in statistics and everyday decision-making. When dealing with a jar containing different colored marbles, calculating the probability of drawing a particular color helps in predicting outcomes, making informed choices, and solving related mathematical problems. In this article, we explore a scenario where a jar contains blue, red, and green marbles with specified probabilities. We will analyze the given probability expressions, find the value of the unknown parameter k, and discuss the implications of these probabilities in practical contexts.
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Overview of the Marble Probabilities
In our scenario, a jar contains marbles of three different colors: blue, red, and green. The probabilities associated with drawing each color are provided as follows:
- Blue marbles: 2/7
- Red marbles: 3/k
- Green marbles: 5/2k
These probabilities are essential because they describe how likely it is to randomly select a marble of a specific color from the jar.
Understanding Probability
Probability measures the chance of an event occurring, expressed as a number between 0 and 1. When dealing with multiple mutually exclusive outcomes, the sum of their probabilities must equal 1, representing the certainty that one of these outcomes will occur.
In this context:
- The probability of drawing a blue marble is 2/7.
- The probability of drawing a red marble is 3/k.
- The probability of drawing a green marble is 5/2k.
Since these are the only possible outcomes (assuming the jar contains only blue, red, and green marbles), the sum of their probabilities must be 1.
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Deriving the Value of k
Given that the probabilities sum to 1, we can set up an equation:
(Probability of Blue) + (Probability of Red) + (Probability of Green) = 1
Substituting the given values:
2/7 + 3/k + 5/2k = 1
Our goal is to find the value of k that satisfies this equation.
Step-by-Step Solution
Step 1: Find a common denominator
To combine the fractions, we need a common denominator. The denominators are 7, k, and 2k. The least common denominator (LCD) is 14k.
Step 2: Rewrite each fraction with the LCD
- 2/7 becomes (2 2k)/(14k) = (4k)/(14k)
- 3/k becomes (3 14)/(14k) = (42)/(14k)
- 5/2k becomes (5 7)/(14k) = (35)/(14k)
Step 3: Write the sum with common denominator
\[
\frac{4k}{14k} + \frac{42}{14k} + \frac{35}{14k} = 1
\]
Simplify numerator:
\[
\frac{4k + 42 + 35}{14k} = 1
\]
Step 4: Cross-multiplied equation
Multiply both sides by 14k:
\[
4k + 42 + 35 = 14k
\]
Combine like terms:
\[
4k + 77 = 14k
\]
Step 5: Solve for k
Subtract 4k from both sides:
\[
77 = 14k - 4k
\]
\[
77 = 10k
\]
Divide both sides by 10:
\[
k = \frac{77}{10} = 7.7
\]
Result: The value of k is 7.7.
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Implications of the Calculated Probabilities
Knowing the value of k allows us to compute the exact probabilities of red and green marbles:
- Red: 3/k = 3/7.7 ≈ 0.3896
- Green: 5/2k = 5/(2 7.7) = 5/15.4 ≈ 0.3247
And the blue probability remains:
- Blue: 2/7 ≈ 0.2857
Adding these probabilities:
\[
0.2857 + 0.3896 + 0.3247 ≈ 0.9999
\]
Due to rounding, this is essentially 1, confirming our calculations are consistent.
Practical implications:
- Predicting outcomes: If you randomly pick a marble from the jar, there's approximately a 28.6% chance it will be blue, 38.9% chance it will be red, and 32.5% chance it will be green.
- Designing experiments: These probabilities can help in designing experiments or games involving marbles.
- Quality control: Understanding the distribution helps in manufacturing or quality assessments if marbles are produced in specific ratios.
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Further Analysis and Applications
Understanding probability distributions like this has broad applications beyond just marbles. Here are some areas where similar concepts are employed:
1. Quality Control in Manufacturing
Manufacturers often analyze the proportion of defective, acceptable, and excellent products in a batch. Probabilities help in assessing the quality and making decisions about process improvements.
2. Gambling and Gaming
In games of chance, probabilities determine payout ratios and game fairness. For example, understanding the odds of drawing certain cards or symbols guides both players and game designers.
3. Risk Assessment and Management
Industries assess the likelihood of various risks (e.g., machine failure, financial loss) to develop mitigation strategies. Probabilities inform policy decisions and resource allocation.
4. Statistical Sampling and Surveys
Sampling techniques rely on probabilities to ensure representative data collection, leading to accurate inferences about larger populations.
Additional Considerations in Probability Problems
When working with probabilities, especially involving unknown parameters like k, several key points should be kept in mind:
- Sum of probabilities: Always verify that the total probability sums to 1, which serves as a consistency check.
- Validity of probabilities: Each probability must be between 0 and 1.
- Parameter constraints: When probabilities involve parameters like k, solving for their values must produce valid probabilities within the [0,1] interval.
- Real-world relevance: Consider whether the theoretical probabilities reflect realistic scenarios.
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Conclusion
Analyzing the problem of a jar containing blue, red, and green marbles with given probabilities demonstrates the importance of algebraic techniques and probability theory in solving real-world problems. By setting up equations based on the total probability rule, we derived the value of k as 7.7, enabling us to determine the exact probabilities of drawing each color. These insights can be applied in various contexts, from manufacturing quality control to game design and risk assessment. Mastery of such probability calculations enhances analytical skills and supports informed decision-making across numerous fields.
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Summary of Key Points
- The probabilities of blue, red, and green marbles are 2/7, 3/k, and 5/2k respectively.
- Assuming these are the only outcomes, their sum must be 1, leading to the equation: 2/7 + 3/k + 5/2k = 1.
- Solving for k yields k = 7.7.
- Calculated probabilities are approximately 28.6% (blue), 38.9% (red), and 32.5% (green).
- Understanding these probabilities allows for better predictions and decision-making in relevant scenarios.
By mastering the process of deriving unknown parameters from probability distributions, students and professionals can confidently analyze similar problems and apply these concepts in practical situations.
