Essential of Probability
assignment week 10
1 Introduction
This week’s mission (aka Weekly Task #10) is all about diving deep and summarizing the absolute essentials of Probability, straight from the supporting video materials in Chapter 6. Getting a solid grip on probability isn’t just nice—it’s mega crucial for statistics, as it sets the foundation for making smart statistical inferences and nailing decisions when things are fuzzy (you know, in the face of uncertainty).
This doc is packing a full-on, comprehensive breakdown of the concepts covered in that video series. The goal? To seriously level up your understanding of how probability theory actually works, making sure you’re totally prepped for the statistical heavy-lifting that’s coming up next.
2 Fundamental Concept
Probability is defined as the chance of an event occurring, calculated by dividing favorable outcomes by total possible outcomes. Understanding this concept is essential for analyzing various events effectively. Calculating probabilities for outcomes when tossing two coins involves identifying possible outcomes and multiplying their probabilities. The sample space reveals four outcomes: HH, HT, TH, and TT.
Understanding probability requires knowing how to calculate the likelihood of events occurring. Key rules include the range of probabilities and the complementary rule, which is essential for solving problems.Understanding probabilities can be achieved through various methods, including the complement rule. This video demonstrates how to calculate the probability of not getting two tails using different approaches.
2.1 Key Points:
- Experiment: Any action whose final outcome cannot be predicted with certainty, such as rolling a die or flipping a coin.
- Sample Space (S): The complete set of all possible outcomes of an experiment. For example, the sample space for rolling a single die is:
- Event (E): A subset of the sample space; one or more outcomes of interest.
- Classical Probability: The probability of an event E occurring is calculated by comparing the number of outcomes that support the event to the total number of all possible outcomes.
- Complement Rule: The probability of event A not occurring is 1 minus the probability of event A occurring. This is very useful for calculating the probability of “at least one” event.
3 Independent and Dependent
Independent events are events in probability that do not affect each other; the outcome of one event has no influence on the outcome of another. The video illustrates this using simple examples such as rolling a die and flipping a coin—two actions that remain completely independent regardless of the result. To calculate the probability of independent events, we multiply the probabilities of each event, following the formula P(A and B) = P(A) × P(B). This formula is essential for determining joint probabilities when events occur independently.
In contrast, dependent events occur when the outcome of one event influences the probability of another. The video explains this concept using an example of drawing marbles without replacement. When an item is removed from the set, the total number of possible outcomes changes, which means the probability for the next draw must be adjusted. This highlights why it is crucial to recognize dependent events—using formulas for independent events in these situations would lead to incorrect results.
To calculate the probability of dependent events, we multiply the probability of the first event by the adjusted probability of the second event, reflecting the updated number of remaining outcomes. Drawing items without replacement is a classic scenario that demonstrates how dependent events work. Ultimately, the main difference between the two concepts lies in whether the outcome of one event affects another. Independent events remain unaffected by each other, while dependent events require continuous adjustment of probabilities after each outcome.
3.1 Key Points:
- Independent Event: The occurrence of A does not affect P(B).
- Dependent Event: The occurrence of A affects P(B).
- Conditional Probability: The probability that B will occur, given that A has occurred.
- Multiplication Formula for Dependents:
4 Union of Events
The video explains the concept of the probability of the union of events. It begins with a review of basic ideas from previous lessons. First, it revisits the definition of a sample space—the complete set of outcomes in an experiment. For example, rolling one six-sided die has six possible outcomes (1–6), while rolling two dice creates 36 total outcomes, which can be arranged into a 6×6 grid.
Next, the video reviews simple probability, defined as the number of favorable outcomes divided by the number of outcomes in the sample space. For instance, the probability of rolling two 4s is 1 out of 36. Additional examples show how to calculate the probability of rolling two even numbers (9/36) and the probability of getting at least one 2 (11/36). The video also connects to the idea of independent events, such as calculating the probability of rolling two 6s using
The lesson then shifts to more complex scenarios. It explains why you cannot multiply probabilities when the events overlap, such as finding the probability of rolling two even numbers and at least one 2. Instead, you must look at the intersection of outcomes, which contains 5 possibilities (5/36).
Finally, the video introduces the formula for the union of events—used when the question contains the word or.
This subtraction term removes duplicate outcomes that belong to both events. A Venn diagram helps visualize how event A (9/36) and event B (11/36) overlap in 5/36 of the sample space. Applying the formula gives the union probability:
The video concludes by reinforcing how the union formula prevents counting overlapping outcomes twice, ensuring accurate probability calculations.
5 Exclusive and Exhaustive
The video introduces mutually exclusive events and exhaustive events using the sample space of rolling two six-sided dice. The sample space consists of all 36 possible outcomes. Mutually exclusive events are defined as events that have no outcomes in common—meaning their Venn diagram circles do not overlap, and the probability of both events occurring together is zero. To illustrate this, the video uses Event A (rolling at least one 5) and Event B (getting a sum less than 4). Event A has 11 favorable outcomes, while Event B has 3. Since the two sets share no outcomes, they are mutually exclusive, and the probability of A and B happening together is 0.
Next, the video explains exhaustive events, which are events that together cover the entire sample space. When represented in a Venn diagram, all outcomes in the box must be highlighted. Using the same dice context, Event A is defined as rolling at least one 6 (11 outcomes), and Event B as getting a sum less than 11 (33 outcomes). Since every outcome is highlighted between A and B, these events are exhaustive. By applying the union formula
the result equals 1, confirming that these events are exhaustive.
The video also highlights that exhaustive events may or may not overlap. In some cases, events can be both mutually exclusive and exhaustive at the same time. For example, Event A (sum is even) and Event B (sum is odd) split the entire sample space into two equal parts. They do not overlap (mutually exclusive), yet together cover all outcomes (exhaustive). Using the union formula again yields a probability of 1, confirming this. The video closes with reminders to support future content and explore additional study resources.
6 Binomial Experiment
A binomial experiment is a statistical experiment used to calculate the probability of obtaining a certain number of “successes” in a fixed number of trials. An experiment is considered binomial if it satisfies four conditions:
- Fixed number of trials (n)
- Each trial has only two outcomes: success or
failure
- Probability of success (p) remains constant
- Trials are independent
These rules allow us to model real-life scenarios such as coin flips, product testing, and any process with two possible outcomes.
6.1 Formula: Combination
The number of ways to choose x successes out of n trials is given by:
\[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \]
6.2 Formula: Binomial Probability
The probability of obtaining x successes in n trials is:
\[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \]
6.3 Summary Table
| Topic | Explanation |
|---|---|
| Definition | Counts number of successes in fixed trials |
| Outcome Type | Success / Failure |
| Condition 1 | Fixed number of trials |
| Condition 2 | Two outcomes |
| Condition 3 | Constant probability (p) |
| Condition 4 | Independent trials |
| Combination Formula | n! / (x! (n-x)!) |
| Binomial Formula | C(n,x) * p^x * (1-p)^(n-x) |
7 Binomial Distribution
Extended Summary
The binomial distribution describes the probability of observing a
certain number of successes (k) in a fixed number of
trials (n), where each trial has only two outcomes
(success/failure), the probability of success (p)
remains constant, and each trial is independent.
It is commonly used in quality control, medical testing, survey
sampling, and any repeated experiment with a yes/no outcome.
7.1 Key Points
| Concept | Explanation |
|---|---|
| Fixed number of trials (n) | Each experiment repeats the same number of times |
| Two outcomes | Success (1) or failure (0) |
| Constant probability (p) | Probability of success does not change |
| Independent trials | One trial does not affect another |
| PMF | Gives the probability of exactly k successes |
| Expected value | Mean = np |
| Variance | σ² = np(1-p) |
7.2 Combination
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
7.3 PMF (Probability of exactly k successes)
\[ P(X = k) = \binom{n}{k}p^k(1-p)^{n-k} \]
7.4 Expected Value
\[ \mu = np \]
7.5 Variance
\[ \sigma^2 = np(1-p) \]
7.6 Visualization 1: Plotting Binomial PMF
# Load library
library(ggplot2)
# Parameters
n <- 10 # number of trials
p <- 0.4 # probability of success
# Values of k
k <- 0:n
# Binomial probabilities
prob <- dbinom(k, size = n, prob = p)
# Data frame
df <- data.frame(k, prob)
# Plot PMF
ggplot(df, aes(x = k, y = prob)) +
geom_bar(stat = "identity") +
labs(
title = "Binomial Probability Mass Function (PMF)",
x = "Number of Successes (k)",
y = "P(X = k)"
) +
theme_minimal()