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Determine the total number of marbles in a jar with blue, black, and green ones, with probabilities of selecting blue and black marbles established, alongside the count of green marbles.
The probability of selecting a blue marble at random from a jar that contains only blue, black and green marbles is 1/5. The probability of selecting a black marble at random from the same jar is 1/4. If the jar contains 11 green marbles, Find the total number of marbles in the jar.
The total number of marbles in the jar is 20.
Given:
- Probability of selecting a blue marble: 1/5
- Probability of selecting a black marble: 1/4
- Number of green marbles: 11
Let’s denote:
- B as the number of blue marbles
- K as the number of black marbles
- G as the number of green marbles
We have the following equations based on probabilities:
- B / (B + K + G) = 1/5 (Probability of selecting a blue marble)
- K / (B + K + G) = 1/4 (Probability of selecting a black marble)
Also, the total number of marbles is given by:
B + K + G = Total number of marbles
Substituting the value of G, we have:
B + K + 11 = Total number of marbles
Now, solving the system of equations:
From equation 1: B = (1/5)(B + K + 11) 5B = B + K + 11 4B = K + 11
From equation 2: K = (1/4)(B + K + 11) 4K = B + K + 11
Substituting K = B into the second equation: 4K = 4B – 11 + 11 4K = 4B
This implies that K = B.
Substituting K = B into 4B = K + 11: 4B = B + 11 3B = 11 B = 11/3
As the number of marbles must be a whole number, we need to reconsider the approach.
Given probabilities:
- P(Blue) = 1/5
- P(Black) = 1/4
Denote the total number of marbles as T. We know: B / T = 1/5 K / T = 1/4 G = 11
Therefore, T = B + K + 11.
Solving for B and K: B = T/5 K = T/4
Substitute B and K into T = B + K + 11: T = (T/5) + (T/4) + 11
Multiply every term by the least common multiple of 5 and 4, which is 20:
20T = 4T + 5T + 220
20T = 9T + 220
11T = 220
T = 220/11
T = 20
Thus, the total number of marbles in the jar is 20.
What is Probability Theory?
Probability theory is a branch of mathematics that deals with quantifying uncertainty and predicting the likelihood of events occurring in various situations. It provides a framework for analyzing random phenomena and making informed decisions in the presence of uncertainty.
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At its core, probability theory seeks to formalize and understand concepts related to chance and randomness. It involves the study of random variables, which are variables whose values are subject to uncertainty, and the assignment of probabilities to different outcomes of a random variable.
Key concepts in probability theory include:
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Sample space: This refers to the set of all possible outcomes of a random experiment.
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Event: An event is a subset of the sample space, representing a particular outcome or a collection of outcomes.
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Probability: Probability quantifies the likelihood of an event occurring and is typically represented as a number between 0 and 1. A probability of 0 indicates impossibility, while a probability of 1 indicates certainty.
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Probability distribution: This describes the probabilities of all possible outcomes of a random variable. It may be discrete, where outcomes can be counted (e.g., rolling a die), or continuous, where outcomes form a range (e.g., measuring the height of people).
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Random variables: Random variables are variables whose values are determined by the outcome of a random phenomenon. They can be discrete or continuous.
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Probability functions: Functions that assign probabilities to events or outcomes of random variables. For discrete random variables, this is often represented by probability mass functions (PMFs), while for continuous random variables, it is represented by probability density functions (PDFs).
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Expected value: Also known as the mean, it represents the long-term average of a random variable over repeated experiments and is a measure of central tendency.
Probability theory finds applications in various fields such as statistics, physics, finance, computer science, and engineering. It is fundamental to understanding uncertainty and making informed decisions in complex systems.
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