GATE CSEIT/Engineering Mathematics/ Conditional probability ( With short trick).

GATE CSEIT/Engineering Mathematics/ Conditional probability ( With short trick).

​Conditional probability is a fundamental concept in probability theory, especially relevant for the GATE Computer Science and Information Technology (CSEIT) exam. It measures the likelihood of an event occurring given that another event has already occurred.​

Definition:

If $A$ and $B$ are two events, the conditional probability of $A$ given $B$ is defined as: P(A∣B)=P(A∩B)P(B)P(A|B) = \frac{P(A \cap B)}{P(B)}P(A∣B)=P(B)P(A∩B)​ provided P(B)≠0P(B) \neq 0P(B)=0.​

Shortcut Techniques for GATE Problems:

1. Utilize Symmetry: In problems where outcomes are symmetrical, symmetry can simplify calculations. For example, when dealing with fair coins or unbiased dice, each outcome has an equal probability, allowing for straightforward computations.

2. Complementary Counting: Sometimes, it’s easier to calculate the probability of the complement of an event and subtract from 1: P(A)=1−P(A′)P(A) = 1 – P(A’)P(A)=1−P(A′) This approach is particularly useful when the event’s complement has a simpler or more direct calculation.

3. Bayes’ Theorem: For problems involving reverse conditional probabilities, Bayes’ Theorem is invaluable: P(A∣B)=P(B∣A)⋅P(A)P(B)P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}P(A∣B)=P(B)P(B∣A)⋅P(A)​ This theorem allows the computation of $P(A|B)$ using $P(B|A)$, which might be more straightforward to determine.

4. Total Probability Theorem: When an event can occur through multiple mutually exclusive scenarios, the total probability is the sum of the probabilities of each scenario: P(B)=∑iP(B∣Ai)⋅P(Ai)P(B) = \sum_{i} P(B|A_i) \cdot P(A_i)P(B)=∑i​P(B∣Ai​)⋅P(Ai​) This technique is useful for breaking down complex problems into manageable parts.

Practice Resources:

• GeeksforGeeks: Offers quizzes and tutorials on conditional probability and Bayes’ Theorem tailored for GATE aspirants. ​

• GATE Overflow: Provides a compilation of previous GATE questions on probability, allowing for targeted practice. ​

• IIT Hyderabad: Hosts a collection of GATE problems in probability, useful for understanding the application of concepts.

Example Problem:

​An urn contains 5 red balls and 5 black balls. A ball is drawn at random and discarded without observing its color. Then, another ball is drawn. What is the probability that the second ball is red?​

Solution:

Since the first ball’s color is unknown and discarded, the total number of balls remains 9 for the second draw. The probability of drawing a red ball in the second draw is: P(Red)=Number of red ballsTotal number of remaining balls=59P(\text{Red}) = \frac{\text{Number of red balls}}{\text{Total number of remaining balls}} = \frac{5}{9}P(Red)=Total number of remaining ballsNumber of red balls​=95​​

Understanding and applying these shortcut techniques can enhance problem-solving efficiency in the GATE exam.

GATE CSEIT/Engineering Mathematics/ Conditional probability ( With short trick).

Conditional Probability