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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

Standard Random Experiments

The following standard random experiments are explained.

•  tossing a coin

•  rolling a dice

•  picking a ball of a color from a box

In each of these, the statistical methods to predict the outcome of an experiment requires large effort and small variations in data can cause errors in the prediction. So the possible outcomes are theorized as having equal chances and the probability of an outcome is defined.

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In the topic Probability of an Event in Random Experiments, we have studied the following.

Dropping a ball is not a random experiment. If a ball is dropped 10 times, it will always hit the floor.

Tossing a coin is a random experiment.

Let us say, when the coin was tossed once, it landed 'heads' up. The result does not indicate in anyway that the result will be 'heads' whenever the coin is tossed. It can either be 'heads' or 'tails'. That is why tossing-a-coin is called a random experiment.

That is, a random experiment has a number of possible outcomes. A random experiment, when performed, results in one of the possible outcomes.

In this chapter, the following random experiments are considered

•  tossing a coin : A coin with 'heads' or 'tails' is flipped. The face seen on top is the outcome of the experiment.

•  rolling a dice : A dice has 6 faces marked with numbers 1 to 6. The dice lands with one of the faces up. The outcome is the number on the face that is facing up.

•  picking a ball from a box : A box has a number of red and blue balls. A person picks a ball without looking into the box. The color of the ball is the outcome of the experiment.

Consider flipping a coin. From the knowledge we have gained in statistics, what is the right way to predict the outcome?

• conduct the experiment a large number of times and collect data. Based on the data one can predict.
• conduct the experiment a large number of times and collect data. Based on the data one can predict.
• the outcome of a random experiment cannot be predicted, so it is not possible.

The answer is "conduct the experiment a large number of times and collect data. Based on the data one can predict."

Considering tossing of a coin. We have conducted the experiment 40 times and collected the data. Which of the following conclusions are correct?

• If the coin is tossed 80 times, the outcome 'heads' will occur 40 times
• If the coin is tossed 20 times, the outcome 'heads' will occur 10 times
• If the coin is tossed 1 time, the outcome 'heads' will be predicted as 20 of 40 times chance.
• all the above
• all the above

The answer is "all the above". This is the most important part of the learning.

Considering tossing a coin. The figure shows examples of conducting the experiment many times. There is small variation in the data What does the data prove?

• In random experiments, the data will have errors due to variations
• In random experiments, the data will have errors due to variations
• Such variation is not possible if a coin is tossed

The answer is "In random experiments, the data will have errors due to variations"

Considering tossing a coin. The figure shows examples of conducting the experiment many times. There is small variation in the data Since, the data has errors due to variations, what does one do?

• nothing further can be done
• the two outcomes are theorized as having equal chance and the data is used to verify if that assumption is correct
• the two outcomes are theorized as having equal chance and the data is used to verify if that assumption is correct

The answer is "the two outcomes are theorized as having equal chance and the data is used to verify if that assumption is correct"

To find the probability of a possible outcome in a random experiment, two possible methods are used.

•  repeat the experiment a large number of times such that any error due to variations is negligibly low.

•  theoretically figure-out that the possible outcomes have equal chance and calculate the probability accordingly.

Probability = number of outcomes in the desired event / number of outcomes in the sample-space.

Considering tossing a coin.

The figure shows examples of conducting the experiment many times. There is small variation in the data Theoretically, the following is derived.

•  the sample space is 'heads' and 'tails'

•  The two outcomes in the sample space have equal chance

•  It is expected that if the experiment is conducted large number of trials, then the difference between number of 'heads' and 'tails' will be negligibly small.

•  Therefore, probability of 'heads' is 1/2 and so for the 'tails'

What does the above prove?

• probability of 'heads' is 1//2
• probability of 'tails' is 1//2
• both the above
• both the above

The answer is "both the above"

A coin that adheres to this theorization of equal probability is called an honest coin.

It is assumed that coins used in random experiments are honest coins. If that assumption is not valid, then it is explicitly given that the coin is biased.

Consider the random-experiment: Rolling a dice. Without conducting the experiment and looking at the data that has errors because of variations, we conclude the following.

•  the sample space is 1, 2, 3, 4, 5, and 6

•  all the six outcomes in the sample space have equal chance.

•  so the probability of 1 is 1//6 and so on for other events.

Which of the following is correct?

• probability of 4 facing up is 1//6
• probability of 4 facing up is 1//6
• probability of 4 facing up is 1//4
• both the above

The answer is "probability of 4 facing up is 1//6"

A dice adhering to this theorization of equal probability is called an honest dice.

Consider the random experiment: Picking a ball at random from a box of 3 red balls and 2 blue balls. Without conducting the experiment and looking at the data that has errors because of variations, we conclude the following

•  The sample space is 'red1', 'red2', 'red3', 'blue1', and 'blue2' color balls.

•  The five balls have equal chance.

•  The desired outcome or the event is a red ball. That can be one of 3 in the sample space.

•  The probability of picking a red ball = count of possible outcomes in desired event / count of possible outcomes in the sample space
Probability of picking a red ball = 3 // 5

What is the probability of picking a blue ball?

• 1//2
• 3//5
• 2//5
• 2//5

The answer is "2//5"

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