CAT exam is all about how smartly you solve the question in the quickest way. To help you with that, we at EduRev have started a summary sheet of formulae which will have all the shortcuts & formulae in one sheet. In this document you would be revising all the important parts of Probability, this document helps you in understanding the shortcuts & formulae of probability in a faster way.
Important Tips & Formulae of Probability
The probability of an event A is the number of ways event A can occur divided by the total number of possible outcomes. The probability of an event A, symbolized by P(A), is a number between 0 and 1, inclusive, that measures the likelihood of an event in the following way:
- If P(A) > P(B) then event A is more likely to occur than event B.
- If P(A) = P(B) then events A and B are equally likely to occur.
- Two events are mutually exclusive if the intersection of two events is null (A ∩ B = φ)
- If E and F two events then P(E U F) = P(E) + P(F) - P(E and F together)
- If an event E is sure to occur, you say that the probability of event E is equal to 1 and you write P (E) = 1. Such events are known as certain events.
- If an event E is sure not to occur, you say that the probability of the Event E is equal to 0 and you write P(E) = 0. Such events are known as impossible events.
- The probability of E not occurring, denoted by P(not E), is given by P(not E) or P
- The odds in favor of occurrence of the event A are defined by m: n – m i.e., P (A): P () and the odds against the occurrence of A are defined by n – m : m, i.e., P (): P (A)
- To find the probability of two or more independent events occurring in sequence, find the probability of each event occurring separately, and then multiply the answers.
Example: If a number is selected at random from the 2 digit 6 multiples, what is the probability that it is divisible by 9?
Solution. Total 6 multiples = 12, - - - -, 96 ⇒ 15.
Required numbers are both multiples of 6 & 9
i. e, the multiples of 18. Those are 18, 36,- - - -, 90, total 5.
∴ Ans =
Question for Examples: Probability
Try yourself:Out of 13 applicants for a job, there are 5 women and 8 men. Two persons are to be selected for the job. The probability that at least one of the selected persons will be a woman is:
The required probability will be given by
First is a woman and Second is a man OR
First is a man and Second is a woman OR
First is a woman and Second is a woman
i.e. (5/13) X (8/12) + (8/13) X (5/12) + (5/13) X (4/12) = 100/156 = 25/39
Alternatively, you can define the non-event as: There are two men and no women. Then, probability of the non- event is
(8/13) X (7/12) = 56/156
Hence, P(E) = (1– 56/156) = 100/156 = 25/39
[Note: This is a case of probability calculation where rep- etition is not allowed.]
Example: If 2 dies are thrown, what is the probability of getting both the numbers prime?
Solution: P (Prime on first die) × P(Prime on second die)
Example: If 3 dies are thrown, what is the probability of getting a sum 4.
Solution: The possibility is 1, 1, 2 or 1, 2, 1 or 2, 1, 1.
∴ Ans =
Example: If 2 cards are drawn from a pack, what is the probability that both the cards are of different colours?
Solution: The possibility is 1 black and 1 Red.
Example: There are 5 red, 4 green, 3 yellow & 8 white balls in a bag, If three balls are chosen at random without replacement, what is the probability that they are of same color.
Solution: All the three can be red or green or yellow or white.
Example: In a game, there are three rounds; the probabilities of winning first, second & third rounds are respectively. A prize will be given, if any one wins all the rounds. What is the probability of winning the prize?
Solution: All the rounds have to be won to get the prize. So, the required answer is.