Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Probability and Statistics → Probability Distributions ↓


Binomial Distribution


The binomial distribution is a fundamental probability distribution in the field of probability and statistics. It is often used to model the number of successes in a given number of independent Bernoulli trials, each of which has the same probability of success. Before getting into its intricacies, let us understand some basic concepts and work on them step-by-step.

Basic terms and concepts

Bernoulli trial

A Bernoulli trial is an experiment or procedure that results in a binary outcome – success or failure. Examples of Bernoulli trials include tossing a coin (where heads can be defined as success and tails as failure) or determining whether or not a light bulb is functional (either it works or it doesn’t).

Independent testing

When trials are independent, the outcome of one trial does not affect the outcome of another trial. Tossing a coin multiple times, assuming the coin is fair, is an example where each toss does not affect the next.

Chances of success

The probability of success is denoted by "p". For a Bernoulli trial such as a coin toss, if the coin is fair, the probability of getting heads (success) is 0.5.

Number of trials (n)

This represents the total number of Bernoulli trials. If you flip a coin 5 times, n = 5.

Number of successes (k)

This is the imaginary number of successes whose probability we want to calculate. For example, if we want to know the probability of getting heads 3 times when we flip 5 coins, then the value of k would be 3.

Interpretation of binomial distribution

In the binomial distribution, we are interested in finding the probability of getting exactly k successes in n independent Bernoulli trials, each of which has a probability of success p. The binomial distribution formula is given as:

P(X = k) = C(n, k) * p^k * (1-p)^(n-k)

Here, C(n, k) is a combination formula, also known as "n choose k", and is calculated as follows:

C(n, k) = n! / (k!(n-k)!)

Where:

  • n! (n factorial) is the product of all positive integers up to n.
  • k! (k factorial) is the product of all positive integers up to k.
  • The factor (n-k)! is the factorial of (n-k).

Real-world examples

Example 1: Tossing a coin

Suppose you flip a fair coin 4 times. What is the probability that you get heads exactly 2 times?

  • Number of trials, n = 4
  • Probability of success, p = 0.5
  • Number of successes, k = 2

The probability is calculated as follows:

P(X = 2) = C(4, 2) * (0.5)^2 * (1-0.5)^(4-2)

Calculating C(4, 2) :

C(4, 2) = 4! / (2!(4-2)!) = (4 * 3) / (2 * 1) = 6

Thus,

P(X = 2) = 6 * 0.25 * 0.25 = 0.375

Therefore, the probability of getting heads exactly 2 times when a fair coin is tossed 4 times is 37.5%.

Example 2: Quality control in manufacturing

A factory makes light bulbs with a 2% defect rate. If a quality control inspector picks 10 bulbs at random, what is the probability that exactly 1 bulb is defective?

  • Number of trials, n = 10
  • Probability of success (faulty), p = 0.02
  • Number of successes, k = 1

The probability is calculated as follows:

P(X = 1) = C(10, 1) * (0.02)^1 * (1-0.02)^(10-1)

Calculating C(10, 1) :

C(10, 1) = 10! / (1!(10-1)!) = 10

Thus,

P(X = 1) = 10 * 0.02 * 0.98^9 ≈ 0.169746

Therefore, there is about a 16.97% chance that exactly 1 out of 10 bulbs will be defective.

Visual representation

To better understand the binomial distribution, consider SVG graph showing the probability distribution of getting k successes in n trials with different probabilities.

0 1 2 3 4 0 0.1 0.2 0.3

In this graph, the height of each bar represents the probability of getting 0 to 4 successes in 4 attempts, with a probability of 0.5 of success for each attempt.

Properties of binomial distribution

Understanding the fundamental properties of the binomial distribution can deepen your understanding and help you apply it in different scenarios:

Mean and variance

  • The mean (average number of successes) for the binomial distribution is given by μ = n * p.
  • The variance (measure of dispersion) for the binomial distribution is given by σ² = n * p * (1 - p).

Symmetry and skewness

The binomial distribution can be symmetric or skewed depending on the value of p:

  • If p = 0.5, then the distribution is symmetric.
  • If p < 0.5, then the distribution is negatively skewed.
  • If p > 0.5, then the distribution is positively skewed.

Relation to other distributions

The binomial distribution is a discrete probability distribution, closely related to the normal distribution. As n (the number of trials) becomes very large, the binomial distribution approaches a normal distribution, which is an important aspect of the central limit theorem.

Conclusion

The binomial distribution, as a discrete probability distribution, is highly useful for modeling binary outcomes in many areas, such as quality control, finance, and any scenario involving success/failure in a series of tests. Its explicit formula and straightforward properties make it an important tool in understanding probability and statistical inference.

Although this is just the beginning of exploring the binomial distribution, this basic understanding provides a foundation upon which more complex statistical models can be built. The key is to recognize its applications and understand how fair and equitable evaluation of Bernoulli trials can provide important insights in both scientific research and practical applications.


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Binomial Distribution

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