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Find out how likely it is that 5 out of 15 adults randomly picked from a certain state didn’t finish high school, based on a recent survey where 81% graduated.
According to a recent survey, 81 percent of adults in a certain state have graduated from high school. If 15 adults from the state are selected at random, What is the probability that 5 of them have not graduated from high school?
The probability that exactly 5 out of 15 adults selected at random have not graduated from high school is approximately 0.0904.
Given:
- p = 0.81 (probability of an adult graduated from high school)
- q = 1 – p = 0.19 (probability of an adult not graduated from high school)
- n = 15 (total adults selected)
- We want to find the probability that 5 out of the 15 selected adults have not graduated from high school.
Using the binomial probability formula:
P(X = 5) = (15 choose 5) * (0.19)^5 * (0.81)^10
Calculating:
(15 choose 5) = 3003 (0.19)^5 = 0.0000301 (0.81)^10 = 0.107374182
Now, multiply these values:
P(X = 5) = 3003 * 0.0000301 * 0.107374182
P(X = 5) ≈ 0.09039
So, the probability that exactly 5 out of 15 adults selected at random have not graduated from high school is approximately 0.09039 or 9.039%.
What is Probability Theory?
Probability theory is a branch of mathematics that deals with the study of random phenomena or events. It provides a framework for quantifying uncertainty and making predictions based on probabilities. The central concept in probability theory is the notion of probability, which measures the likelihood of a particular event occurring.
Key components of probability theory include:
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Sample Space: The set of all possible outcomes of a random experiment.
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Event: A subset of the sample space, representing one or more outcomes of interest.
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Probability Function: A function that assigns a numerical value between 0 and 1 to each event, representing the likelihood of that event occurring. This function satisfies certain axioms, such as the probability of the sample space being 1 and the probability of the empty set being 0.
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Probability Distribution: A mathematical function that describes the probabilities of different outcomes in a random experiment.
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Random Variables: Variables that can take on different values according to the outcomes of a random experiment. Each value has an associated probability.
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Joint Probability: The probability of two or more events occurring simultaneously.
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Conditional Probability: The probability of an event occurring given that another event has already occurred.
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Independence: Two events are independent if the occurrence of one event does not affect the probability of the other event occurring.
Probability theory finds applications in various fields such as statistics, physics, finance, engineering, and computer science. It is essential for making informed decisions in situations involving uncertainty and randomness.
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