Probability

Ramesh and Raina are two friends, who are fond of solving math problems. The teacher gave them an equation and told them that whoever solves the equation first, will give them a gift. The boys started to solve the equation. Now, the chance of Ramesh and Raina getting the gift is fifty-fifty. So, we can say that the probability of Ramesh or Raina getting the gift is 1/2. 

When a person that the possibility of happening something is high, then we can assume that the case of it happening is affirmative and vice versa. For instance, compare it with the probability of Rain. 

Therefore, the above-mentioned cases are all examples of Probability. The probability meaning in Hindi is “sambhavna”, meaning Chance. Changes in a case happening are never predicted and certain. There is no certain answer to the question of “who will earn the gift- Ramesh or Raina?” and “will it rain heavy or there will be no rain?” There is uncertainty because the information we have is not complete. This is the reason behind the usage of Probability in different fields. The probability of a case happening cannot be more than 1 or less than 0. It must lie between these two numbers. 

Now, we know what is probability, let us discuss the probability formula. 

Probability Formula

The basic probability formula is simple and easy to remember. The formula is-

Number of assenting outcomeTotal outcome (or Sample Space)

Therefore,

P (X)= n (X)S

Since probability is the occurrence of a case. We can say that the case is the subset of the total Sample Space.

That is,

X ⸦ S

In addition to this, other formulas are used along with this. Let us assume that there are two cases- Q and F

• The probability formula for Conditional Probability is

P( QF ) = P (Q Ո F)P (F),

OR

P( FQ ) = P (Q Ո F)P (Q).

• The probability formula for Independent Cases is 

P (Q Ո F) = P (Q) . P (F).

• The addition rule is 

P (Q Ս F) = P (Q) + P (F) – P (Q Ո F).

• Bayes’ Formula is 

P( QF ) = P (FQ) . P (Q)P (F).

The above-mentioned probability formulas are used according to the needs of the questions. Otherwise, we use the basic Probability formula. 

Related terms

The following important concepts are often used along with the probability formula to calculate the uncertainty.

1. Mutually exclusive cases are the events that happen whenever two different cases cannot end up happening at the same time. That is, the outcome for the occurrence of these cases is different. Consider a coin, it has two sides- heads and tail. It is practically not possible for this coin to give an outcome, which has is both head and tail. Therefore, it is a mutually exclusive case.
2. A case where the probable outcomes are foreseen, however, the exact probability is not known is widely known as a Random Experiment. For instance, when we toss a fair coin, the probable outcomes are- Head or Tail. However, we don’t know if a Head or a tail will show up. Here, all the probable outcomes are the Sample Space of the case.
3. The sum of all the smaller cases of probability is known as the Total Probability. The total probability can never be greater than 1, or less than 0.
4. There are instances where the possibility is not dependent on the occurrence and non-occurrence of a case. These instances are known as Independent Cases. For instance, Rajiv and Rahul selected two pairs of shoes, each. The probability of Rajiv buying a pair of shoes is independent of the probability of Rahul’s choices.
5. The chances of a case having more than one case are known as Compound cases. The chances of each of the outcomes are equally possible. Let us understand this with the help of the example of rain, that we have used earlier. However, there can be other outcomes for this- there may be a thunderstorm, a hurricane, etc.