Examples: Probability

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 =

Practice Question

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:

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.
i.e,

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.

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## FAQs on Examples: Probability - CSAT Preparation - UPSC

 1. What is probability theory?
Ans. Probability theory is a branch of mathematics that deals with the study of uncertainty and randomness. It provides a framework for understanding and quantifying the likelihood of events occurring. This theory is used extensively in various fields, including statistics, economics, physics, and computer science.
 2. What is the difference between theoretical probability and experimental probability?
Ans. Theoretical probability is based on mathematical calculations and is determined by analyzing the possible outcomes of an event. It is often expressed as a fraction or percentage. On the other hand, experimental probability is calculated based on actual observations or experiments. It involves conducting trials and recording the outcomes to estimate the probability of an event occurring.
 3. How can probability be applied in real-life situations?
Ans. Probability theory has numerous practical applications in everyday life. It can be used to analyze weather forecasts, predict the likelihood of a stock market crash, determine the chances of winning a lottery, or assess the risk of certain diseases. Additionally, it helps in making informed decisions in fields such as insurance, gambling, and risk management.
 4. What are the basic principles of probability theory?
Ans. The basic principles of probability theory include the Law of Large Numbers, which states that as the number of trials increases, the experimental probability approaches the theoretical probability. Additionally, the Addition and Multiplication Rules help determine the probability of compound events. The Complement Rule allows calculating the probability of an event not occurring, and the Conditional Probability Rule deals with events dependent on previous outcomes.
 5. How can probability be used to analyze data?
Ans. Probability plays a crucial role in data analysis and statistical inference. By applying probability distributions, such as the normal distribution, researchers can assess the likelihood of certain values occurring and make inferences about a population based on sample data. Probability also helps in hypothesis testing, where the probability of obtaining certain results under the null hypothesis is evaluated to make conclusions about the population.

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