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Probabilty

Understand the terms related to probability

Observing an Experiment

It is not always possible to tell the exact outcome of a particular action. Take, for example, a dart board.

A dart is repeatedly thrown toward the dartboard, targeting a random number in each throw. We do not know which number is targeted in a particular throw. What we do know is that there is a fixed group of numbers and each time the targeted number is one of them.

We know that the likelihood of occurrence of an unpredictable event is studied under the theory of probability. So, we can say that there is a certain probability for each number to be targeted in the above experiment.

Let us learn more about probability and the meanings of terms associated with it, for example, ‘experiment’ and ‘outcome’.

Did You Know?

The word ‘probability’ has evolved from the Latin word ‘probabilitas’, which can be considered to have the same meaning as the word ‘probity’. In olden days in Europe, ‘probity’ was a measure of authority of a witness in a legal case, and it often correlated with the nobility of the witness.

The modern meaning of probability, however, focuses on the statistical observation of the likelihood of occurrence of an event.

Know More

Probability is widely applicable in daily life and in researches pertaining to different fields. It is an important factor in the diverse worlds of share market, philosophy, artificial intelligence or machine learning, statistics, etc. All gambling is based on probability. In gambling, one considers all possibilities and then tries to predict a result that is most likely to happen. The concept of probability is perhaps the most interesting topic to discuss in mathematics.

Terms Related to Probability

Experiment: When an operation is planned and done under controlled conditions, it is known as an experiment. For example, tossing a coin, throwing a die, drawing a card from a pack of playing cards without seeing, etc., are all experiments. A chance experiment is one in which the result is unknown or not predetermined.

Outcomes: Different results obtained in an experiment are known as outcomes. For example, on tossing a coin, if the result is a head, then the outcome is a head; if the result is a tail, then the outcome is a tail.

Random: An experiment is random if it is done without any conscious decision. For example, drawing a card from a well-shuffled pack of playing cards is a random experiment if it is done without seeing the card or figuring it out by touching.

Trial: A trial is an action or an experiment that results in one or several outcomes. For example, if a coin is tossed five times, then each toss of the coin is called a trial.

Sample space: The set of all possible outcomes of an experiment is called the sample space. It is denoted by the English letter ‘S’ or Greek letter ‘Ω’ (omega). In the experiment of tossing a coin, there are only two possible outcomes—a head (H) and a tail (T).

∴ Sample space (S) = {H, T}

Event: The event of an experiment is one or more outcomes of the experiment. For example, tossing a coin and getting a head or a tail is an event. Throwing a die and getting a face marked with an odd number (i.e., 1, 3 or 5) or an even number (2, 4 or 6) is also an event.

Know More

Initially, the word ‘probable’ meant the same as the word ‘approvable’ and was used in the same sense to support or approve of opinions and actions. Any action described as ‘probable’ was considered the most likely and sensible action to be taken by a rational and sensible person.

Whiz Kid

Equally Likely: If each outcome of an experiment has the same probability of occurring, then the outcomes are said to be equally likely outcomes.

Girolamo Cardano (1501−1576) was a great Italian mathematician, physicist, astrologer and gambler. His interest in gambling led him to do more research on the concept of probability and formulate its rules. He was often short of money and kept himself solvent through his gambling skills. He was also a very good chess player. He wrote a book named Liber de Ludo Aleae. In this book about games of chance, he propounded the basic concepts of probability.

Solved Examples

Easy

Example 1:

A fair die is thrown. What is the sample space of this experiment?

Solution:

When a die is thrown, we can have six outcomes, namely, 1, 2, 3, 4, 5 and 6.

We know that sample space is the collection of all possible outcomes of an experiment.

∴ Sample space (S) = {1, 2, 3, 4, 5, 6}

Example 2:

Which of the following are experiments?

i)Tossing a coin

ii)Rolling a six-sided die

iii)Getting a head on a tossed coin

Solution:

Tossing a coin and rolling a six-sided die are experiments, while getting a head on a tossed coin is the outcome of an experiment.

Medium

Example 1:

What is the sample space when two coins are tossed together?

Solution:

When two coins are tossed together, we can get four possible outcomes. These are as follows:

i)A head (H) on one coin and a tail (T) on the other

ii)A head (H) on one coin and a head (H) on the other

iii)A tail (T) on one coin and a head (H) on the other

iv)A tail (T) on one coin and a tail (T) on the other

∴ Sample space (S) = {HT, HH, TH, TT}

Suppose there are two pens of different colors in a box. One is blue and the other is black. If you draw a pen from the box, without looking at the colour, then can you say that it will be the blue pen or the black pen surely?

You can never say with surety whether the pen will be of blue color or black color. Thus, there are two possible outcomes and there is an equal chance of both the possible outcomes to occur.

Since the two possible outcomes have equal chances of occurrence, the chance for each outcome to occur is half. Thus, the probability of drawing a blue pen or a black pen from the box will be same and equal to.

Now, consider the case that the box contains one blue pen and two black pens.

Here, the probability of drawing a black pen will be more than the probability of drawing a blue pen. This is because the number of black pens is more than the number of blue pens in the box.

Further, the number of black pens is twice the number of blue pens. Thus, the probability of drawing a black pen is twice the probability of drawing a blue pen.

The probability of drawing a blue pen is and the probability of drawing a black pen is.

Let us consider one more example to understand the concept better.

Sanjana throws a dice. She can get any of the numbers: 1, 2, 3, 4, 5, or 6 on the top face of the dice. There is no other possibility. Moreover, there is an equal chance of getting these numbers, i.e., the probability of getting any of these numbers is the same. Since the number of possible outcomes is 6, the probability of getting any of these numbers is.

Now, when she throws a dice, can she get the number 7 on the top face of the dice?

Observe that when a dice is thrown, then we cannot get the number 7. Thus, the probability of getting the number 7 is nothing but zero.

Thus, the probability of the event which has no possibility to occur is 0.

Also, observe that the probability of getting any of the numbers 1, 2, 3, 4, 5, and 6 on the top face of the dice is 1 as we will surely get any one of the numbers from 1 to 6 upon throwing a die.

Thus, we can say that the probability of the event which is sure to occur is always 1.

From the above examples, we conclude the following facts:

1) The probability of occurrence of any event always lies between 0 and 1.

2) The probability of such an event which has no possibility to occur is 0.

3) The probability of such an event which is sure to occur is 1.

Let us look at some more examples now.

Example 1:

When a coin is tossed, what is the probability of getting a head?

Solution:

When a coin is tossed, there are two possible outcomes − head or tail. The possibility of occurrence of both of them will be the same. Therefore, the probability of getting a head is equal to the probability of getting a tail. Thus, the probability of getting a head is equal to.

Example 2:

A dice is thrown. What is the probability of getting

the number 2 any one of the number among 1, 2, 3, 4, 5, and 6 the number 7

Solution:

When a dice is thrown, there are six possible outcomes: 1, 2, 3, 4

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