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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

### Probability of an Event in Random Experiments

In this page, learn how statistical data can be used to predict an outcome of a random experiment.

Such prediction requires large effort and small variations in the data can cause errors.

So the possible outcomes are theorized as having equal chances and the probability of an outcome is defined.

click on the content to continue..

A person is tossing a coin for 40 times, and recording the data. The data is shown in tabular form for 4 different trials. If the coin is tossed once, what is the probability of 'tails'? Is it 21//40, 18//40, 20//40, or 19//40?

• the data is more or less 1//2, so the probability = 1//2
• the data is more or less 1//2, so the probability = 1//2
• the data is not uniform, so the probability cannot be measured.

The answer is "the data is more or less 1//2, so the probability = 1//2".

This is not a precise explanation, though broadly it is correct. The next page provides the precise explanation for taking the probability as 1//2.

A person is tossing a coin for 40 times, and recording the data. The data is shown in tabular form for 4 trials. The nature of random experiments is that, there is always some variation in outcome. The occurrence of 'tails' is 21//40, 18//40, 20//40, or 19//40 in four trials.

But, we understand that the 'heads' and 'tails' have equal chance and so we theorize that the probability of 'tails' is 1/2. The data from experiments is used to verify if the result of the experiment is approximately 1/2.

Note: We started with collecting data and representing that.

Then we learned using data to predict result of random experiments.

Now, as the data is not accurate, (that is data varies little in small number of experiments), we try to learn parameters of the experiment and make a call on predicting the results. In such prediction, data helps to validate the prediction. This is further explained with examples in pages to follow.

Consider tossing a coin.

What is the sample-space?

• 'heads' and 'tails'
• 'heads' and 'tails'
• flipping a coin

The answer is, 'heads' and 'tails'

Consider tossing a coin. The sample space is 'heads' and 'tails'. Do these have equal chance in the experiment?

• No. 'tails' occurs more often
• Yes. The result can be either one with equal chance.
• Yes. The result can be either one with equal chance.

The answer is "Yes. The result can be either one with equal chance".

Consider tossing a coin. The sample space is 'heads' and 'tails' and all in the sample space has equal chance. What is the probability of 'heads'?

• Since the 2 outcomes have equal chance, probability is 1//2
• Since the 2 outcomes have equal chance, probability is 1//2
• cannot find the probability without collecting data

The answer is "Since the 2 outcomes have equal chance, probability is 1//2", where

•  numerator 1 is the count of outcomes in the event

•  denominator 2 is the count of outcomes in the sample-space.

The number of outcomes in the sample space is n.

The outcomes in sample space have equal chance to be the result.

The desired event is one of p outcomes.

The probability = p//n.

In tossing a coin for 'heads':

•  The sample space is 'heads' and 'tails'. That is a count of 2.

•  The desired event is the outcome 'heads'. That is a count of 1.

•  The Probability of event 'heads' = text(count of desired outcome)/text(count of sample-space) =1/2.

Consider rolling a dice. What is the sample-space?

• rolling a dice
• '1', '2', '3', '4', '5', and '6'
• '1', '2', '3', '4', '5', and '6'

The answer is "'1', '2', '3', '4', '5', and '6'"

Consider rolling a dice. The sample space is '1', '2', '3', '4', '5', and '6'. Do these have equal chance in the experiment?

• No. The outcomes do not have equal chance.
• Yes. The outcomes have equal chance.
• Yes. The outcomes have equal chance.

The answer is "Yes. The outcomes have equal chance".

Consider rolling a dice. The sample space is '1', '2', '3', '4', '5', and '6' and all outcomes in the sample space have equal chance.

What is the probability of '4'?

• Since the six outcomes have equal chance, probability of one outcome is 1//6
• Since the six outcomes have equal chance, probability of one outcome is 1//6
• cannot find the probability without collecting data

The answer is "Since the six outcomes have equal chance, probability of one outcome is 1//6", where

•  numerator 1 is the count of outcomes in the event

•  denominator 6 is the count of outcomes in the sample-space.

The number of outcomes in the sample space is n.

The outcomes in the sample space have equal chance.

The desired event is one of p outcomes.

The probability = p//n.

In rolling a dice for 4:

•  The sample space is '1', '2', '3', '4', '5', and '6'. That is a count of 6.

•  The desired event is the outcome '4'. That is a count of 1.

•  The Probability of event '4' = text(count of desired outcome)/text(count of sample-space) =1/6.

Consider rolling a dice. The sample space is '1', '2', '3', '4', '5', and '6' and all outcomes in the sample space have equal chance.

What is the probability of an odd number?

• 1//6
• 3//6
• 3//6

The answer is "3//6". Since the six outcomes have equal chance, probability of three outcomes is 3//6. where

•  numerator 3 is the count of outcomes in the event (1, 3, 5)

•  denominator 6 is the count of outcomes in the sample-space.

In rolling a dice for an odd number:

•  The sample space is '1', '2', '3', '4', '5', and '6'. That is a count of 6.

•  The desired event is an odd number which is the outcomes '1', '3', or '5'. That is a count of 3.

•  The Probability of event "odd number" = text(count of desired outcome)/text(count of sample-space) = 3/6.

Probability in Equal Chance Experiments : The number of outcomes in the sample space is n.

The outcomes in sample space have equal chance.

The desired event is one of p outcomes.

The probability = p//n.

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