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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, then is it a random experiment?

• Yes. The process of recording is done and so it is random experiment.
• No. Since we already know the result, it is not random.
• No. Since we already know the result, it is not random.

The answer is "No. Since we already know the result, it is not random."

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. Is this toss a random experiment?

• Yes. The first toss does not affect the result of the second toss.
• Yes. The first toss does not affect the result of the second toss.
• No. We already know the result as 'tails'.

The answer is "Yes. The first toss does not affect the outcome of the second toss".

Which of the following is a meaning for the word "random"?

• unknown; not defined; chance for many possibilities
• unknown; not defined; chance for many possibilities
• running to a dome on mountain

The answer is "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.

Which of the following is an outcome?

• 3 glasses of water
• 3 glasses of water
• 20 students

The answer 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".

Which of the following is an outcome?

• tossing a coin
• heads
• heads

The answer is "heads".

Which of the following is a meaning for the word "outcome"?

• to come out of a building
• result; something produced or created
• result; something produced or created

The answer is "result".

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".

What is the sample space for the experiment?

• 1, 2, 3, 4, and 5 glasses
• 1, 2, 3, 4, and 5 glasses
• only 1, 2, and 3

The answer is "1, 2, 3, 4, and 5 glasses". The sample-space 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, it did not happen.

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".

What is the sample space for the experiment?

• 'heads' and 'tails'
• 'heads' and 'tails'
• only 'heads'

The answer is, 'heads' and 'tails'.

Which of the following is a meaning for the phrasae "sample-space"?

• extent or stretch of the possibilities
• extent or stretch of the possibilities
• a rectangle cut into small pieces

The answer is "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".

Which of the following is a sample point?

• 3 glasses of water
• 3 glasses of water
• 20 students

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

Which of the following is a sample-point?

• tossing a coin
• heads
• heads

The answer is "heads".

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. Which of the following is the event for that?

• one sample point -- 2 glasses of water -- is the event
• one sample point -- 2 glasses of water -- is the event
• one sample point cannot be an event

The answer is "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'. Which of the following is the event?

• 'heads' and 'tails'
• only 'tails'
• only 'tails'

The answer is "tails".

Which of the following is a meaning for the word "event"?

• a gathering to celebrate birthday
• something that happens or takes place
• something that happens or takes place

The answer is "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.

What is the probability of 1 glass?

• 11//40
• 11 times out of every 40 times
• both the above
• both the above

The answer is "both the above". Note: The probability is given as a fraction.

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.

What is the probability of 'heads'?

• 20//40
• 1//2
• both the above
• both the above

The answer is "both the above". 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.

Which of the following is a meaning for the word "probability"?

• round and adorable
• measure of likelihood of an event
• measure of likelihood of an event

The answer is "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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