Probability – Revision of Definitions
- Probability is a measure of how likely an event is to occur.
- Sample space is the set of all possible outcomes of an experiment.
- Event is a subset of a sample space.
- Experiment is a procedure that can produce a set of outcomes.
- Random variable is a function that assigns a numerical value to each outcome in the sample space.
Probability – Basic Concepts
- Probability ranges from 0 to 1, where 0 indicates an impossible event and 1 indicates a certain event.
- The probability of an event can be determined using the formula:
P(E) = Number of favorable outcomes / Total number of outcomes
- The complement of an event A is denoted by A’, which represents all outcomes that are not in A.
- The complement rule states that
P(A') = 1 - P(A)
Probability – Laws and Rules
- Addition Law: For disjoint events A and B,
P(A or B) = P(A) + P(B)
- Multiplication Law: For independent events A and B,
P(A and B) = P(A) * P(B)
- Conditional Probability: The probability of an event A given that another event B has already occurred is denoted by
P(A | B) and calculated using P(A | B) = P(A and B) / P(B)
Probability – Examples
- Example 1: If a fair die is rolled, find the probability of getting a prime number.
- Sample space = {1, 2, 3, 4, 5, 6}
- Favorable outcomes = {2, 3, 5}
- Probability = 3/6 = 1/2
- Example 2: Two cards are drawn successively without replacement from a well-shuffled deck of 52 cards. Find the probability of getting two red cards.
- Sample space = 52 cards
- Number of red cards = 26 (half of the deck)
- Probability = (26/52) * (25/51) = 25/102
Probability – Conditional Probability
- Conditional probability is the probability of an event occurring given that another event has already occurred.
- It is denoted by P(A | B), where A and B are two events.
- Conditional probability is calculated using the formula:
P(A | B) = P(A and B) / P(B)
Probability – Bayes’ Theorem
- Bayes’ theorem is used to calculate conditional probabilities when the order of events is reversed.
- It is stated as:
P(A | B) = (P(B | A) * P(A)) / P(B)
Probability – Bayes’ Theorem Example
- Example: A disease test is 95% accurate. The probability of having the disease is 2%. What is the probability of a positive test result indicating the presence of the disease?
- Let A be the event of having the disease and B be the event of a positive test result.
- P(A) = 2% = 0.02
- P(B | A) = 95% = 0.95
- P(B) = ?
- Using Bayes’ theorem: P(A | B) = (P(B | A) * P(A)) / P(B)
Probability – Expected Value
- The expected value of a random variable is a measure of its centrality.
- Expected value is denoted by E(X).
- It is calculated by multiplying each possible value of the random variable by its probability and summing them up.
Probability – Expected Value Example
- Example: A fair die is rolled. Let X be the random variable representing the outcome. Find the expected value of X.
- X can take values from 1 to 6, each with a probability of 1/6.
- Expected value: E(X) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = 3.5
Probability – Variance and Standard Deviation
- Variance measures the spread of values around the expected value.
- Standard deviation is the square root of variance and also measures the spread of values.
- Variance is denoted by Var(X) and standard deviation is denoted by σ(X).
- Variance is calculated using the formula:
Var(X) = E(X - μ)^2
- Standard deviation is calculated as the square root of variance:
σ(X) = √Var(X)
Probability – Permutations and Combinations
- Permutations: The arrangement of objects where the order matters.
- Permutation formula:
nPr = n! / (n - r)! (n = total number of objects, r = number of objects selected)
- Combinations: The selection of objects where the order does not matter.
- Combination formula:
nCr = n! / (r! * (n - r)!) (n = total number of objects, r = number of objects selected)
Probability – Permutations and Combinations Example
- Example 1: In how many ways can 3 people be seated in a row of 6 chairs?
- Permutations (order matters): 6P3 = 6! / (6 - 3)! = 6! / 3! = (6 * 5 * 4) = 120 ways
- Combinations (order does not matter): 6C3 = 6! / (3! * (6 - 3)!) = (6 * 5 * 4) / (3 * 2 * 1) = 20 ways
Probability – Binomial Theorem
- The binomial theorem provides a formula for expanding expressions of the form
(a + b)^n
- It is stated as:
(a + b)^n = nC0 * a^n + nC1 * a^(n-1) * b + nC2 * a^(n-2) * b^2 + ... + nCr * a^(n-r) * b^r + ... + nCn * b^n
Probability – Binomial Theorem Example
- Example: Expand
(x + 2)^4
- Using the binomial theorem, we have:
(x + 2)^4 = 4C0 * x^4 + 4C1 * x^3 * 2 + 4C2 * x^2 * 2^2 + 4C3 * x * 2^3 + 4C4 * 2^4
- Simplifying, we get:
x^4 + 8x^3 + 24x^2 + 32x + 16
Probability – Random Variables
- Random variables are variables whose values are determined by the outcomes of a random experiment.
- Discrete random variables take on distinct values.
- Continuous random variables can take on any value within a given range.
- Probability distribution is a function that assigns probabilities to the possible values of a random variable.
Probability – Probability Distribution
- Probability distribution provides the probabilities associated with each possible value of a random variable.
- It can be presented in tabular or graphical form.
- For discrete random variables, probability distribution can be represented as a probability mass function.
- For continuous random variables, probability distribution can be represented as a probability density function.
Probability – Probability Distribution Example
- Example: Consider the rolling of a fair six-sided die. Let X be the random variable representing the outcome.
- The probability distribution of X is:
- X = 1 with probability 1/6
- X = 2 with probability 1/6
- X = 3 with probability 1/6
- X = 4 with probability 1/6
- X = 5 with probability 1/6
- X = 6 with probability 1/6
Probability – Mean of a Random Variable
- The mean of a random variable is the expected value of the variable.
- For a discrete random variable, it is calculated by summing the product of each possible value and its probability.
- For a continuous random variable, it is calculated by integrating the product of each value and its probability density function.
Probability – Mean of a Random Variable Example
- Example: Consider the rolling of a fair six-sided die. Let X be the random variable representing the outcome.
- Mean (expected value) of X = E(X) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = 3.5
Probability – Variance and Standard Deviation of a Random Variable
- Variance measures the spread of values around the mean.
- Standard deviation is the square root of variance and also measures the spread of values.
- Variance is calculated using the formula:
Var(X) = E((X - μ)^2)
- Standard deviation is calculated as the square root of variance:
σ(X) = √Var(X)
Probability – Continuous Probability Distributions
- Continuous probability distributions are used to model random variables that can take on any value within a given range.
- The probability of obtaining a single value is zero, and probabilities are described using probability density functions (PDFs).
- PDFs are non-negative functions and integrate to 1 over the entire range of the random variable.
Probability – Normal Distribution
- The normal distribution is a continuous probability distribution that is widely used to model real-world phenomena.
- It is symmetric and bell-shaped, with the mean, median, and mode all equal.
- The standard normal distribution has a mean of 0 and a standard deviation of 1.
- The probability density function of the standard normal distribution is given by: 
Probability – Standard Normal Distribution
- The standard normal distribution has a mean of 0 and a standard deviation of 1.
- Z-scores or standard scores are used to measure deviations from the mean in terms of standard deviations.
- Z-scores are calculated using the formula:
Z = (X - μ) / σ, where X is the variable, μ is the mean, and σ is the standard deviation.
Probability – Standardizing a Normal Distribution
- Standardizing a normal distribution involves transforming it into a standard normal distribution.
- This is done by calculating the Z-score for each value using the formula:
Z = (X - μ) / σ
- Once standardized, probabilities can be calculated using the standard normal distribution table or calculator.
Probability – Normal Distribution Example
- Example: The heights of adult males follow a normal distribution with a mean of 70 inches and a standard deviation of 3 inches. What is the probability of selecting a male with a height between 68 inches and 72 inches?
- We need to standardize the values using the Z-score formula:
Z = (X - μ) / σ
- For 68 inches:
Z = (68 - 70) / 3 = -2/3
- For 72 inches:
Z = (72 - 70) / 3 = 2/3
- Using the standard normal distribution table, we can find the probabilities associated with these Z-scores.
Probability – Poisson Distribution
- The Poisson distribution is used to model the number of events occurring in a fixed interval of time or space.
- It is characterized by a single parameter λ (lambda), which represents the average rate of occurrence of the events.
- The probability mass function of the Poisson distribution is given by: 
Probability – Poisson Distribution Example
- Example: The number of cars passing through a particular intersection per hour follows a Poisson distribution with an average rate of 10 cars per hour. What is the probability that exactly 3 cars will pass through the intersection in a given hour?
- Using the Poisson distribution formula: P(x; λ) = (e^(-λ) * λ^x) / x!, where x = 3 and λ = 10
- With x = 3 and λ = 10, we can calculate the probability using the formula.
Probability – Binomial Distribution
- The binomial distribution is used to model the number of successes in a fixed number of independent Bernoulli trials.
- It is characterized by two parameters: n, the number of trials, and p, the probability of success in each trial.
- The probability mass function of the binomial distribution is given by: 
Probability – Binomial Distribution Example
- Example: A fair coin is tossed 10 times. What is the probability of getting exactly 5 heads?
- Using the binomial distribution formula: P(x; n, p) = (nCx) * (p^x) * ((1-p)^(n-x)), where x = 5, n = 10, and p = 0.5 (since it is a fair coin)
- With x = 5, n = 10, and p = 0.5, we can calculate the probability using the formula.
Probability – Conclusion
- Probability is a fundamental concept in mathematics and statistics that allows us to quantify uncertainty.
- Through the study of probability, we can make predictions, solve real-world problems, and inform decision-making.
- It is important to understand the basic concepts, laws, and rules of probability, as well as the different probability distributions.
- By mastering these concepts, you will be able to apply probability principles effectively in various situations.