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Learn the probability of randomly selecting 2 balls, with one being white and the other blue, out of a mixed bag containing 3 red, 6 white, and 7 blue balls.
A bag contains 3 red, 6 white and 7 blue balls. What is the probability that 2 balls drawn are white and blue?
The probability that 2 balls drawn are white and blue is 7/20.
There are two ways to get a white ball and a blue ball: either a white ball first, then a blue ball, or vice versa. So, the probability is the sum of these two probabilities.
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Probability of white ball, then blue ball:
- There are 6 white balls, and we draw one, leaving 5 white balls remaining.
- There are 7 blue balls, and we draw one, leaving 6 blue balls remaining.
- The total number of balls is 16, and we’ve drawn 1, so there are 15 balls remaining.
Therefore, the probability of drawing a white ball, then a blue ball is:
(Number of white balls) / (Total number of balls) * (Number of blue balls remaining) / (Remaining balls after drawing white) = 6/16 * 7/15 = 7/40
Probability of blue ball, then white ball:
- There are 7 blue balls, and we draw one, leaving 6 blue balls remaining.
- There are 6 white balls, and we draw one, leaving 5 white balls remaining.
- The total number of balls is 16, and we’ve drawn 1, so there are 15 balls remaining.
Therefore, the probability of drawing a blue ball, then a white ball is:
(Number of blue balls) / (Total number of balls) * (Number of white balls remaining) / (Remaining balls after drawing blue) = 7/16 * 6/15 = 7/40
Total probability:
The probability of getting a white ball and a blue ball, regardless of the order, is the sum of these two probabilities:
7/40 + 7/40 = 14/40 = 7/20
Therefore, the probability that 2 balls drawn are white and blue is 7/20.
Applications of Probability in Real World
Probability theory has numerous applications in various real-world contexts. Here are some examples:
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Risk Assessment and Insurance: Insurance companies heavily rely on probability theory to assess risks and set premiums. Actuaries use probability models to predict the likelihood of events such as accidents, illnesses, or natural disasters occurring to determine appropriate insurance rates.
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Finance and Investments: Probability plays a crucial role in financial markets for risk management, portfolio optimization, and pricing of financial derivatives. Traders and investors use probability models to make informed decisions about buying, selling, or holding financial assets.
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Weather Forecasting: Meteorologists use probabilistic models to forecast weather conditions. By analyzing historical weather data and current atmospheric conditions, they estimate the likelihood of various weather outcomes, such as rain, snow, or temperature changes.
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Medical Diagnosis and Treatment: In healthcare, probability theory is used in medical diagnosis, treatment planning, and clinical trials. Physicians assess the probability of diseases based on symptoms, test results, and patient history, guiding their diagnostic and treatment decisions.
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Quality Control and Manufacturing: Probability theory is applied in quality control processes to ensure product reliability and consistency in manufacturing. Techniques such as statistical process control use probability distributions to monitor and improve production processes.
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Game Theory and Decision Making: Probability is fundamental in game theory, which analyzes strategic interactions among rational decision-makers. It helps in understanding outcomes and making optimal decisions in scenarios like negotiations, auctions, and competitive markets.
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Sports Analytics: Probability models are used in sports analytics to evaluate player performance, predict match outcomes, and optimize strategies. For example, in baseball, sabermetrics uses probability-based metrics to assess player contributions and team performance.
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Genetics and Biology: Probability models are used in genetics to analyze inheritance patterns, gene expression, and population genetics. In epidemiology, probabilistic models help in understanding the spread of diseases and designing effective interventions.
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Telecommunications and Networking: Probability theory is used in designing communication systems and networks to analyze data transmission, packet routing, and network reliability. It helps in optimizing system performance and managing network congestion.
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Law and Forensic Science: Probability theory is applied in legal contexts for evidence evaluation, forensic analysis, and courtroom decision-making. Techniques such as Bayesian inference are used to assess the strength of evidence and make judgments in criminal investigations and civil litigation.
These are just a few examples highlighting the broad range of applications of probability theory across various domains in the real world.
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