Probability – Introduction

• Probability is a measure of the likelihood of an event occurring.
• It is used to study and understand uncertain events.
• Probability is expressed as a number between 0 and 1.
• Events with a probability of 0 are impossible, while events with a probability of 1 are certain to occur.

Terminologies in Probability

• Experiment: Any activity or process that generates an outcome.
• Sample Space: The set of all possible outcomes of an experiment, usually denoted by S.
• Event: A subset of the sample space, denoted by E.
• Probability of an Event: The likelihood of an event occurring, denoted by P(E).
• Complementary Event: The event that does not happen, denoted by E'.
• Equally Likely Outcomes: Outcomes that have the same probability of occurring.

Types of Probability

1. Theoretical Probability:
   • Calculated by using theoretical principles.
   • Formula: P(E) = (Number of favorable outcomes) / (Total number of possible outcomes)

2. Experimental Probability:
   • Calculated by conducting experiments and observing outcomes.
   • Formula: P(E) = (Number of times event occurs) / (Total number of experiments)

3. Subjective Probability:
   • Based on personal opinion or judgment.
   • No formula involved, rather relies on an individual’s experience or perception.

Addition Rule of Probability

• The addition rule is used to calculate the probability of the union of two events, denoted by E ∪ F.
• Formula: P(E ∪ F) = P(E) + P(F) - P(E ∩ F)
• P(E ∩ F) represents the probability of both events E and F occurring simultaneously.

Mutually Exclusive Events

• Mutually exclusive events are events that cannot occur at the same time.
• If two events, E and F, are mutually exclusive, P(E ∩ F) = 0.
• In this case, the addition rule simplifies to: P(E ∪ F) = P(E) + P(F).

Conditional Probability

• Conditional probability is the probability of an event occurring given that another event has already occurred.
• Formula: P(E|F) = P(E ∩ F) / P(F)
• P(E|F) denotes the probability of event E occurring given that event F has already occurred.

Independent Events

• Independent events are events that do not affect each other’s probabilities.
• If two events, E and F, are independent, P(E|F) = P(E) and P(F|E) = P(F).
• The formula for calculating the probability of both events is: P(E ∩ F) = P(E) * P(F)

Multiplication Rule of Probability

• The multiplication rule is used to calculate the probability of the intersection of two events, denoted by E ∩ F.
• Formula: P(E ∩ F) = P(E) * P(F|E)

Permutations

• A permutation is an arrangement of objects in a specific order.
• The number of permutations of n objects taken r at a time is given by the formula: nPr = n! / (n-r)!

Combinations

• A combination is a selection of objects where the order does not matter.
• The number of combinations of n objects taken r at a time is given by the formula: nCr = n! / (r!(n-r)!)

Sure! Here are slides 11 to 20 on the topic of Probability for the 12th Boards Math exam.

Remark and Special Cases

• If P(A) = 0, then the event A is considered impossible.
• If P(A) = 1, then the event A is considered certain.
• P(A’) = 1 - P(A), where A’ is the complementary event of A.

Examples:

• If a fair coin is tossed, the probability of getting a head is 0.5.
• If a dice is rolled, the probability of getting a prime number is 2/6 or 1/3.

Problem Solving Strategies

1. Identify the event(s) and their probabilities.

2. Determine the required probability using the appropriate rules.

3. Simplify the expression if possible.

4. Calculate the desired probability.

Example:

• A bag contains 4 red balls and 6 blue balls. If a ball is randomly chosen, what is the probability of selecting a red ball?

Conditional Probability - Examples

1. A card is drawn from a deck of playing cards. What is the probability of drawing a king given that the card drawn is a face card?

2. A bag contains 5 red balls and 3 blue balls. Two balls are drawn without replacement. Find the probability of drawing both red balls.

Solution:

1. P(King|Face Card) = P(King and Face Card) / P(Face Card)

2. P(Red and Red) = P(Red) * P(Red|Red first)

Independent vs. Dependent Events

• In independent events, the outcome of one event does not affect the outcome of the other event.
• In dependent events, the outcome of one event affects the probability of the other event.

Example:

• Rolling a fair die twice: The outcome of the first roll does not affect the outcome of the second roll, making them independent events.

Law of Large Numbers

• The law of large numbers states that as the number of trials or experiments increases, the experimental probability approaches the theoretical probability.

Example:

• Tossing a fair coin - As the number of tosses increases, the probability of getting a head or tail approaches 0.5.

Factorial Notation

• Factorial notation is denoted by an exclamation mark (!) and is used to calculate the number of arrangements or combinations.
• n! represents the product of all positive integers less than or equal to n.
• Examples: 3! = 3 x 2 x 1 = 6, 5! = 5 x 4 x 3 x 2 x 1 = 120

Permutations - Examples

• How many ways can 4 books be arranged on a bookshelf?
• How many ways can the letters of the word “MATH” be arranged?

Solution:

1. The number of permutations of 4 books = 4P4 = 4!

2. The number of permutations of the word “MATH” = 4P4 = 4!

Combinations - Examples

• A committee of 3 members needs to be formed from a group of 10 people. How many possible combinations are there?
• How many ways can 5 students be selected from a group of 20 students to form a team?

Solution:

1. The number of combinations of 3 members from 10 people = 10C3 = 10! / (3! * (10-3)!)

2. The number of combinations of 5 students from 20 students = 20C5 = 20! / (5! * (20-5)!)

Probability Distributions

• Probability distributions show the probabilities of different outcomes in an experiment.
• They are often represented in the form of tables, graphs, or formulas.

Example:

• A fair 6-sided die is rolled. The probability distribution of getting each number from 1 to 6 is equal, which is 1/6.

Summary

• Probability is a measure of the likelihood of an event occurring.
• It can be calculated using theoretical, experimental, or subjective methods.
• The addition rule is used for calculating the probability of the union of two events.
• The multiplication rule is used for calculating the probability of the intersection of two events.
• Permutations and combinations help calculate the number of possible arrangements and selections.

Bayes’ Theorem

• Bayes’ Theorem is used to calculate the probability of an event occurring based on prior knowledge or evidence.
• Formula: P(A|B) = (P(B|A) * P(A)) / P(B)
• P(A|B) denotes the probability of event A occurring given that event B has already occurred.

Example:

• A blood test is conducted to identify a disease. The probability of the test being positive if the person has the disease is 0.95, and the probability of a false positive is 0.03. If 1% of the population has the disease, what is the probability that a person who tests positive actually has the disease?

Random Variables

• A random variable is a variable whose values are determined by the outcome of a random experiment.
• It can take on different values based on the probabilities associated with those values.

Example:

• Rolling a fair die - The random variable represents the possible outcomes (1, 2, 3, 4, 5, or 6) and their associated probabilities (1/6 for each).

Probability Distributions for Random Variables

• Probability distributions for random variables show the probabilities associated with each possible value of the random variable.
• It can be represented in the form of a table, graph, or formula.

Example:

• If a random variable X represents the number of heads obtained when flipping a fair coin twice, the probability distribution is as follows:
  • P(X=0) = 1/4
  • P(X=1) = 1/2
  • P(X=2) = 1/4

Expected Value

• The expected value of a random variable is the long-term average value it takes over many repetitions of the experiment.
• It is calculated by multiplying each possible value of the random variable by its corresponding probability and adding them up.

Formula:

• Expected Value (E(X)) = ∑ (x * P(x)) where x represents each possible value of the random variable, and P(x) represents the probability of that value.

Example:

• If a random variable X represents the number obtained when rolling a fair die, the expected value is:
  • E(X) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = 3.5

Variance and Standard Deviation

• Variance is a measure of the spread or dispersion of a random variable’s values around its expected value.
• Standard deviation is the square root of the variance and provides a measure of the average amount by which the values deviate from the expected value.

Formulas:

• Variance (Var(X)) = ∑ [(x - E(X))^2 * P(x)]
• Standard Deviation (SD(X)) = √Var(X)

Example:

• If a random variable X represents the number obtained when rolling a fair die, the variance and standard deviation are calculated as follows:
  • Variance = [(1-3.5)^2 * 1/6] + [(2-3.5)^2 * 1/6] + [(3-3.5)^2 * 1/6] + [(4-3.5)^2 * 1/6] + [(5-3.5)^2 * 1/6] + [(6-3.5)^2 * 1/6]
  • Standard Deviation = √Variance

Probability Density Function (PDF)

• The probability density function (PDF) is used to describe the likelihood of a continuous random variable taking on a specific value.
• Unlike discrete random variables, continuous random variables can take any value within a specified range.

Example:

• If the random variable X represents the height of students in a class, the PDF will describe the distribution of the heights across a continuous range, such as from 150 cm to 180 cm.

Cumulative Distribution Function (CDF)

• The cumulative distribution function (CDF) provides the probability that a random variable takes on a value less than or equal to a given value.
• It is defined as the integral of the PDF from negative infinity to the given value.

Example:

• If the random variable X represents the height of students in a class, the CDF will provide the probability that a student’s height is less than or equal to a certain value.

Sampling Distribution

• A sampling distribution is a probability distribution that describes the likelihood of obtaining different sample statistics from a population.
• It helps in understanding the behavior of statistics such as the mean or standard deviation when repeatedly sampling from a population.

Example:

• If a population of heights follows a normal distribution, the sampling distribution of the sample mean height will also tend to be normally distributed.

Central Limit Theorem

• The central limit theorem states that the sampling distribution of the mean of any independent, randomly drawn sample approaches a normal distribution as the sample size increases.
• This holds true regardless of the shape of the population distribution.

Example:

• Suppose a population has a skewed distribution. As the sample size increases, the distribution of the sample mean will approach a normal distribution.

Hypothesis Testing

• Hypothesis testing is a statistical procedure to make inferences about population parameters based on sample data.
• It involves formulating null and alternative hypotheses, calculating test statistics, and determining the significance level to accept or reject the null hypothesis.

Steps:

1. State the null and alternative hypotheses.

2. Set the significance level (α).

3. Calculate the test statistic based on sample data.

4. Compare the test statistic with the critical value(s) from the appropriate distribution.

5. Make a decision to either reject or fail to reject the null hypothesis.