Probability

Probability of Random Event

In every paper of IBPS PO, SO and SBI PO, there will be a few questions on the Probability of Random Event. These questions form the subset of quantitative aptitude section. For both IBPS and SBI the quantitative Aptitude section forms a very important section. Here we shall define the concept of probability. We will also see what do we mean when we call an event a random event. Moreover, we will develop techniques that will help you understand and calculate the probability of such events. Let us begin by understanding the concept of probability.

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The Probability of Random Event

Let us first try and understand the concept of probability. In general sense of the word, the probability of something means the chance of its occurrence or the chances that we will observe an event at a certain time. For example, when someone says that the probability it raining today is high, you understand that they mean that there is a high chance that it will rain.

But what if someone says that the probability of something happening is very low? You understand that the person means that the event may not occur at all. In mathematics, we define probability in a similar sense. Before we write the rules and the formula that defines the probability of an event, let us see what we mean by an event in mathematics.

Event

Consider a simple example. Let us say that we toss a coin up in the air. What can happen when it gets back? It will either give a head or a tail. These two are known as outcomes and the occurrence of an outcome is an event. Thus the event is the outcome of some phenomenon.

                                                                                                                                                     Source: Quora

Now that we know what an event is, let us see the following terms. these terms will be used throughout the article and are crucial to the understanding and solving of the concepts of probability.

Terms related to Probability

  • Random Experiment: A random experiment is one in which all the possible results are known in advance but none of them can be predicted with certainty.
  • Outcome: The result of a random experiment is called an outcome.
  • Sample Space: The set of all the possible outcomes of a random experiment is called Sample Space, and it is denoted by  ‘S’.
  • Event: A subset of the sample space is called an Event.

For example, consider the coin toss again. If we represent the occurrence of the Head by H and the tail by a T, then we can write {H, T} as the sample space. Since there is no other physical possibility, this is the set that contains all the possible outcomes of the event i.e. coin toss.

Definition Of Probability

Now let us introduce the formal definition of the probability of an event. Let us say that for some event E, ‘N’ is the total number of possible outcomes. For example, if E is a coin toss, then N = 2 i.e. H and T. Out of these ‘N’ possible outcomes, let us say we want to find the probability of some event X, that can happen in ‘n’ ways. Then we can write the probability of occurrence of the event X as:

P (X) = (Number Of Ways In Which X Can Happen)/(Total Number Of Ways In Which The Event Can Happen)

Here, P (X) represents the probability of the event X. Thus we can write:

P (X) = n/N; where ‘n’ is the number of the favourable outcomes and ‘N’ is the number of total possible outcomes.

Solved Examples On Probability

Example 1: Find the probability for a randomly chosen month to be January?

Answer: If we choose something out of say ‘n’ things, then the number of favourable outcomes is 1 (since we are choosing only one thing) and the number of unfavourable outcomes is equal to the number of the things that we are choosing from. Here we are choosing one month (January) out of a total of twelve months. So the probability of choosing any month from the given 12 months is = 1/12.

Example 2: Find the probability that a given day chosen in a week is Sunday?

Answer: There are 7 possible days (Monday to Sunday) in a week. So the probability that it will be a Sunday = 1/7.

Example 3: One integer is chosen from 1, 2, 3, … 100. What is the probability that it is neither divisible by 4 nor divisible by 6?

Answer: From numbers 1-100, Numbers divisible by 4 = 25. Numbers divisible by 6 = 16. Numbers divisible by 12 (LCM of 4 and 6) = 8.

Also, Numbers divisible by 4 or 6 = 25 + 16 – 8 = 33. Hence, numbers which are not divisible by 4 or 6 = 100-33 = 67

Therefore the probability that the number chosen, is neither divisible by 4 nor divisible by 6  = 67/100 = 0.67

Notice that the value of the Probability of a random event always lies between 0 and 1. If you get an answer that is not between 0 and 1, it is wrong.

Practice Questions:

Q 1: From a well-shuffled deck of cards, find the probability of drawing a King.

A) 0.0769 units             B) 0.0679 units              C) 0.0769           D) 0.0679

Ans: C) 0.0769

Q 2: An unbiased die is rolled. What is the probability of getting a six?

A) 0.167                 B) 1.67               C) 16.7                   D) 6

Ans: A) 0.167

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