Probability Formulas with Examples for Probability Aptitude Problems | FACE Prep

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In this article, we will discuss the basic concepts of probability, probability formulas, and example problems. This will help you get an overall idea of how to approach any Probability-based questions.

 

Probability Concept

The word probability does not have a direct definition but it can be defined as a measure of the likelihood for the occurrence of some event. In many contexts, the word probability is used synonymously with a chance. Probability of an event always lies between 0 to 1. Here is a picture that will help you understand better.

probability formulas to solve probability problems

Note: The concepts of permutations and combinations are also used in determining the probability of an event.

 

Must know Probability Terms & Definitions

The following definitions are fairly critical in obtaining a complete understanding of the subject.

  • Deterministic experiment: If the outcome of an experiment is certain, then it is called a deterministic experiment.
  • Random experiment: If the outcome of an experiment is not unique and can be one amongst many possible outcomes, then it is called a random experiment. Examples of performing a random experiment are rolling an unbiased dice, Tossing a fair coin, Drawing a card from a pack of well-shuffled cards, Picking up a ball of a certain color from a bag containing balls of different colors.
  • Sample space: The set of all possible outcomes of an experiment is called sample space. Examples of Sample Space are: In tossing a coin, S={ H, T} and In rolling dice, we have S={ 1, 2, 3, 4, 5, 6}.
  • Event: A subset of the sample space is called an event. It comprises of one or more outcomes of the experiment.
  • Biased experiment: If the likelihood of occurrence of one outcome is more than other outcomes, then the experiment is biased.
  • Unbiased experiment: If all the outcomes of the experiment are equally likely to happen, then the experiment is called an unbiased experiment.
  • Mutually exclusive events: Two events are said to be mutually exclusive if the occurrence of one event eliminates the possibility of the occurrence of another. For example, when a coin is tossed once, the occurrence of heads and tails are mutually exclusive as both of them cannot happen simultaneously.
  • Collectively exhaustive events: Events that together cover all possible outcomes are called collectively exhaustive events. For example, when a die is thrown, getting an odd number and getting an even number are two events which put together will account for all possible outcomes. These events are called collectively exhaustive.
  • Independent events: Two events are said to be independent if the occurrence of one event does not affect the occurrence of another. For example, when a coin is tossed and a die is thrown simultaneously, the event of getting a head and the number ‘6’ is an independent event.

Note: If the events are mutually exclusive and collectively exhaustive, then the sum of probabilities of the events will be equal to 1.

 

Probability Formulas and Tricks

Here are some of the important Probability Formulas that you need to understand to solve the problems in no time.

  • When we throw a coin, then either a Head(H) or a Tail (T) appears.
  • Dice is a solid cube having six faces, marked 1, 2, 3, 4, 5, 6 respectively. When we throw a die, the outcome is the number that appears on its upper face.
  • A pack of cards has 52 cards. It has 13 cards of each suit, namely Spades, Clubs, Hearts and Diamonds. Cards of Spades and clubs are black cards. Cards of Diamonds and hearts are red cards. There are four honours of each suit. These are Aces, Kings, Queens and Jacks. These are called face cards.
  • Probability of Occurrence of an event: Let S be the sample space and E be an event. Then, E<= S.

          P(E) = n(E) / n(S)
          P(S) = 1
          0<= P(E) <= 1
          P(Ø) = 0
          For any events A and B , we have P(A U B) = P(A) + P(B) – P(A ∩ B)

  • If an experiment has ‘n’ outcomes in total, out of which ‘m’ outcomes are in favor of one particular event, then the probability of that event is the ratio of m and n. The probability of an event E is denoted by P(E).

          important probability formulas

  • For a given event E, non-occurrence of the event is denoted by the symbol E¯. Events E and E¯ put together are mutually exclusive and collectively exhaustive. If ‘m’ outcomes are favorable to a given event E, then ‘ n-m’ events are favorable for the event E¯, where ‘n’ denotes the total number of outcomes. The probability of the event is denoted by P(E¯) and is called the complementary event for E.       

         probability formulas for probability aptitude questions

  

Theorems of Probability | Probability formulas and tricks

1) Addition Theorem

  • Addition theorem on probability states that for a given set of events A and B with probabilities P(A) and P(B),
  • P (A∪B) = P(A) + P(B) - P (A∩B)
  • If A and B are mutually exhaustive events, then P (A∩B) = P(A) x P(B)
  • If A and B are mutually exclusive events, then P (A∩B) = 0
  • In other words for mutually exclusive events, P (A∪B) = P(A) + P(B)

2) Multiplication Theorem

If A and B are independent events, then P (A∩B) = P(A) x P(B)  i.e. the probability of simultaneous occurrence of independent events is equal to the product of the individual probabilities for the occurrence of the events.

3) Conditional Probability

The Probability Formulas of the occurrence of an event A, given that event B, has already occurred is given by

conditional probability formulas

4) Expected value

If a monetary value is associated with each outcome of an experiment, then the expected value from the experiment, if the experiment is performed multiple times, can be computed as the weighted average of the monetary values for all outcomes of the experiment, with the weights being the individual probabilities of the outcomes.

Probability Formulas can be generalized as follows

probability formulas of expected value

Example:

Let us say that in a gambling house, betting happens on the throw of an unbiased die. We know that the probability of getting any particular number between 1 and 6 (say 5) in this event is 1/6. In the event of a 6, the casino house pays out Rs. 120 to every participant and in the case of any other number, it charges Rs. 30 to every participant. Now, the expected value of the game, if the game is played over a long period of time, can be calculated as

E(G) =  (1/6) x 120 + (5/6) x -30 = -5

Hence, if the game is played a long period of time, the casino will benefit by an amount of Rs. 5 for each experiment.

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