Mathematics for Biotechnology

Sticky Ad Probability Problems with Detailed Solutions Click each question to expand the detailed interpretation and solution. Question 1: Two unbiased dice are thrown Two fair dice are rolled together. Find the probability that: (i) the sum is even (ii) the sum is less than 5 (iii) the sum is at most 12 Since each die has 6 possible outcomes, the total number of outcomes is: $$6\times6=36$$ We calculate the number of favorable outcomes for each condition. (i) Sum is even The sum is even when: both numbers are even both numbers are odd Number of even numbers on a die = 3 Number of odd numbers on a die = 3 $$3\times3+3\times3=18$$ Therefore, $$P(\text{sum is even})=\frac{18}{36}=\frac12$$ (ii) Sum is less than 5 Possible sums are 2, 3 and 4. Number of favorable outcomes: $$1+2+3=6$$ H...

Lecture Notes on Set Theory with Detailed Solutions Lecture Notes on Set Theory Topics: Definition of Set, Types of Sets, Set Operations, Theorems, Addition Theorem, Detailed Solved Examples, Applications in Biology/Biotechnology/Neuroscience Table of Contents Definition of Set Types of Sets Union and Intersection Addition Theorem De Morgan's Laws Solved Examples Applications in Biology / Biotechnology / Neuroscience Practice Exercises Summary 1. Definition of Set 2. Types of Sets 3. Union and Intersection 4. Addition Theorem 5. De Morgan's Laws 6. Solved Examples 7. Applications in Biology / Biotechnology / Neuroscience 8. Practice Exercises 9. Summary 1. Definition of Set A set is a well-defined collection of distinct objects. Example: $A = \{1,2,3,4\}$ is a set of natural numbers. Applications: Genes expressed in a cell can be represented as a set. Neurons activated under a s...