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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.
mathsStatistics and ProbabilityBasics of Probability

Introduction to Random Experiments

The following are introduced and explained

•  random experiment

•  possible outcomes

•  sample space

•  sample point

•  event

•  probability of an event

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Consider the number of glasses of water students drink during normal day, collected for 40 students. Using the data, we try to predict the data for 1 student.

Considering one student to record the data is the "experiment".

In doing the experiment, one chooses the student without any knowledge about the student. Such experiment is "random experiment".

If we choose a student to record data, knowing already that he took 4 glasses water, It is not a random experiment.

Consider tossing a coin, and recording the data for 40 tosses. Using the data, we try to predict the result of 1 toss.

Tossing the coin once is an "experiment".

In doing the experiment, one chooses the coin that does not favour either side. Such experiment is "random experiment".

We tossed the coin once and found the result as 'tails'. Then again, toss the coin. The first toss does not affect the outcome of the second toss, so the second toss is a random experiment.

The word "random" means: unknown; not defined; chance for many possibilities.

Random Experiment : A process or occurrence for which the result cannot be predicted with certainty.

Consider the number of glasses of water students drink during normal day, data collected for 40 students. Using the data, we try to predict the data for 1 student.

The number-of-glasses a student drinks is the "outcome" of the experiment.

For example, in one instance of the random experiment, one student drinks 3 glasses of water. The outcome for this instance is 3 glasses of water.

Consider tossing a coin, and recording the data for 40 tosses. Using the data, we try to predict the data for 1 toss.

The face of the coin (heads or tails) pointing up after the toss is the "outcome".

For example, when a coin is tossed, the face of the coin is heads. Tossing a coin is a random experiment, and the outcome of this trial is "heads".

The word "outcome" means result ; something produced or created.

Outcome : A result of an random experiment is an outcome.

Consider the number of glasses of water students drink during normal day, data collected for 40 students. Using the data, we try to predict the data for 1 student. All possible outcomes are "the sample space".

For example, the sample space of this experiment is 1, 2, 3, 4, or 5 glasses.

Note: The outcome may include 0, 6, or 7 glasses. In such case, the sample space includes 0, 1, 2, 3, 4, 5, 6, and 7 glasses. But in the given particular case, the sample space is smaller.

Consider tossing a coin, and recording the data for 40 tosses. Using the data, we try to predict the data for 1 toss.

All possible outcomes are "the sample space".

The sample space for the given example is 'heads' and 'tails'.

The phrase "sample-space" means: extent/stretch of the possibilities.

Sample Space : The set of all possible outcomes, when the experiment is repeated many times over, is the sample space.

Consider the number of glasses of water students drink during normal day, data collected for 40 students. Using the data, we try to predict the data for 1 student.

One outcome in the sample-space is a "sample point".

For example, 3 glasses of water is a sample point.

Consider tossing a coin, and recording the data for 40 tosses. Using the data, we try to predict the data for 1 toss.

When collecting data, we consider if the outcome is 'heads' or 'tails'. One outcome in the sample space is a "sample point".

For this example, "heads" is a sample point.

Sample Point : One outcome in the sample-space is a sample point.
Outcome and sample point are interchangeably used. Outcome is used in the context of experiment and sample-point is used in the context of sample space.

Consider the number of glasses of water students drink during normal day, data collected for 40 students. Using the data, we try to predict the data for 1 student.

One experiment is to find possibility of 3 or more glasses of water for one student. In that case, the outcomes are 3, 4, and 5. The set of desired outcome together is called an "event".

In an experiment, the desired outcome is 2 glasses of water. In this case, 2 glasses of water is the event.

Consider tossing a coin, and recording the data for 40 tosses. Using the data, we try to predict the data for 1 toss.

One experiment is to find possibility of 'tails'. The outcome "tails" is the event for the specified experiment.

The word "event" means: something that happens or takes place.

Event : A desired outcome or a combination of multiple desired outcomes is an event.

Consider the number of glasses of water students drink during normal day, data collected for 40 students. Using the data, we try to predict the data for 1 student.

One experiment is to find possibility of 2 glasses of water for one student. The prediction is 18 times out of 40 times. The prediction about an event based on data is the "probability" of the event.

The probability of 1 glass is understood as 11 times out of every 40 times and is given as 11/40.

Consider tossing a coin, and recording the data for 40 tosses. Using the data, we try to predict the data for 1 toss.

One experiment is to find the possibility of 'tails' for one toss. The prediction is 20 times out of 40 times. The prediction about an event base on data is the "probability" of the event.

The probability of 'heads' is understood as 20 times out of every 40 times and is given as 20/40 = 1/2.

Note: The fraction representing a probability can be simplified as per the rules of simplification of fractions.

Probability of "heads" in "toss of a coin" is 1/2, which is understood as for a large number of tosses, the event "heads" appears 1 out of 2 times.

Probability 1/2 does not mean for every 2 toss, head appears 1. On the contrary, it is a possibility that if a coin is tossed 3 times, the event "tails" appears all 3 times.

If the coin is tossed a million times, then very close to half a million time the "heads" is expected to appear.

Repeating, the probability 1/2 means, in large number of repetitions, the event appears 1 out of 2 times.

The word "probability" means: measure of likelihood of an event.

Probability : The number of times an event is likely to appear per experiment is the probability of the event.

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