Chance and Probability

Introduction

Chance and Probability are very similar to each other. The concept of experiment has a profound presence, structurally, in the fundamentals of the theory of probability. The main concept of the probabilistic approach in any scientific thought is simply to judge the nature of the experiments through which the thought itself may acquire a strong foothold. This sort of judgment is, however made in terms of the likelihood of something appearing as the result of an experiment.

It is true that an observation or a test is usually initiated with an intense hope that it will be ended up with a prefixed conclusion. Though, on the contrary, most of the experiments barring a few do not come up with results predictable in advance. The result or outcome of each of these experiments is indeed one in many possibilities. If such an experiment i.e., an experiment with unpredictable results are repeated several times under identical situation, their successive results may differ from each other. This type of experiment is termed as random experiment.

In this tutorial, we will learn about Probability, difference between the terms Probability and Possibility.

Probability

We often use, when we converse with others, certain phrases like ‘almost likely’, ‘most probably’, ‘most certainly’ etc. These merely indicate the chance of an event to occur in reference to others. Thus, by using these, we guess whether an event has a better chance of occurrence than that other events. This we commonly do through our intuition and experience. But an experience based pattern of reasoning, in most cases, fails wretchedly to reach the actuality.

Probability measures the chance of occurrence of a favourable event in a random experiment. Measuring anything always needs a unit. Of all the events, sure event S has a hundred percent chance to occur and thus, the probability of S is considered to be the unit; we write it as P(S)=1. The probability of the impossible event ϕ is eventually assumed to be zero P(ϕ)=0. The probability of any other event is a part of P(S). Intuition demands that for any event A, P(A) can never be zero, where P(A) is the property of A.

If you closely analyse the definition, you can easily see that one of the preconditions of applicability of the definition is that the possible outcomes of the experiment must be ‘equally likely’. This clearly shows a shortcoming of the definition in the sense that it is useless in experiments where the outcomes are not ‘equally likely’. For example, if a loaded die is used for throwing, surely the outcomes are no more ‘equally likely’, whence you cannot apply the classical definition to find the probability of ‘getting a six’ in this case.

Indeed, the problem is more critical. Though one may intuitively feel the sense of the term ‘equally likely’, it actually claims that the events are ‘equally probable’, whence the classical definition in seen to define probability in terms of probability itself. This circularity in classical definition is its great weakness.

Possibility vs Probability

Possibility is simply the chance of the occurrence of an event. When we express it mathematically using digits and numbers, then it is called probability. Probability is nothing but the mathematical representation of how likely an event is to occur. It is all about guessing whether an event will occur or not, and if it occurs, then what percent chance it has of occurrence.

Solved Examples

1)What is the chance that a leap year, selected at random, will contain 53 Sundays?

A leap year has 52 complete weeks and 2 more days. These 2 days will be 2 consecutive days or the first and the last day of the week. Then the number of such pairs of days in a week is 7 namely

• Monday and Tuesday

• Tuesday and Wednesday

• Wednesdays and Thursday

• Thursday and Friday

• Friday and Saturday

• Saturday and Sunday

• Sunday and Monday

To get another Sunday in a leap year, we have to select 2 pairs i.e. (6) & (7) out of 7 such pairs. So, the required probability is \frac{2}{7}.

2)A cricket match is played from 10 a.m. to 4 p.m. A spectator arrives to see the match. What is the probability that he will miss the first sixer of the match which takes place within the 40th minute of the match?

The spectator has to reach between 10 a.m. and 4 p.m. So, the total length of the time is equal to 360 minutes. The first sixer takes place between 10 a.m. and 10:40 a.m. So, he will miss the first sixer if he arrives between 10:40 a.m. and 4 p.m. So, the favorable length of time= 5hrs. 20 minutes =320 minutes.

So, the probability of his missing the first sixer $\mathrm{\frac{320}{360}=\frac{8}{9}}$.

Conclusion

In this tutorial, we have learned about Probability, the difference between the terms Probability and Possibility. Probability is nothing but the mathematical representation of how likely an event is to occur. It indicates the chance of an event to occur in reference to others. Thus, by using these, we guess whether an event has a better chance of occurrence than that other events. This we commonly do through our intuition and experience. But an experience-based pattern of reasoning, in most cases, fails wretchedly to reach the actuality.

FAQs

1. Briefly explain what a random event is.

If a sample space S of a random experiment E is discrete, then any subset of S is called a random event of E.

2. Briefly explain what simple event & compound event is.

An event consisting of a single point of the sample space is called a simple or elementary event. An event which is not simple is called a compound event.

3. Briefly explain what discrete and continuous sample space is.

A sample space, which is a finite, or a countably infinite set is called a discrete sample space. A sample space, when an uncountably infinite set, is called a non-discrete or continuous sample space.

4. What is the number of possible outcomes when a die is thrown thrice?

A die has six faces. So, the number of outcomes in each throw of the die is 6. With each of the six outcomes in the first throw, 6 possible outcomes in the second throw will be associated. Thus, the number of possible outcomes in throwing the die twice is 6×6=36. With each of these 36 outcomes, there will be associated 6 possible outcomes in the third throw. So, 36×6=216 is the total number of possible outcomes of the experiment of throwing a die thrice.

5. Briefly explain equally likely events.

Two events are said to be equally likely if none of them can be expected to occur in preference to the other.