Subjects
Biology
Chemistry
AP Chemistry
English
AP English Literature
Arithmetic
Nov 23, 2025
54
4 pages
River Przygoda @riverprzygoda_jvnr
This chapter covers the fundamental concepts of probability, introducing how we calculate the likelihood of events occurring in... Show more
Probability helps us measure how likely events are to happen. The basic formula for finding the probability of an event A is P(A) = n(A)/n(U), which compares the number of favorable outcomes to the total possible outcomes.
When working with multiple events, we need specific formulas. For complementary events (something either happens or doesn't), we use P(A) + P(A') = 1. For combined events, the formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B) accounts for potential overlap.
Mutually exclusive events can't happen at the same time, simplifying our formula to P(A ∪ B) = P(A) + P(B). When events might influence each other, we use conditional probability P(A|B) = P(A ∩ B)/P(B). If events don't affect each other, they're independent, and P(A ∩ B) = P(A)P(B).
💡 Remember that the union symbol (∪) represents "OR" situations, while the intersection symbol (∩) represents "AND" situations when calculating probabilities.
Random experiments are unpredictable processes that don't yield the same results each time, even under identical conditions. The sample space (S) contains all possible outcomes, with each individual outcome called a simple event.
Any subset of the sample space is an event (E). Events can be simple (containing just one outcome) or compound (containing multiple outcomes). For any probability calculation to make sense, probabilities must always be between 0 and 1, and all probabilities in a sample space must add up to 1.
We can determine probability through relative frequency, which is calculated as f(E)/n, where f(E) is how many times an event occurs in n trials. This approach is particularly useful when we have experimental data rather than theoretical probabilities.
🔑 When identifying events, always start by determining the appropriate sample space. For example, a two-child family could have sample space {BB, BG, GB, GG} if birth order matters, or {0, 1, 2} if you're only counting the number of girls.
When working with dice or similar random experiments, we can map out all possible outcomes to find probabilities. For instance, when rolling two dice, there are 36 possible outcomes, and the event "sum equals 5" includes four possibilities (1,4), (4,1), (2,3), and (3,2).
Experimental probabilities come from observed data rather than theoretical calculations. If you flip two coins 1,000 times and HH appears 273 times, you might assign P(HH) = 0.273. From this experimental data, you can calculate compound event probabilities by adding the relevant simple event probabilities.
Finding probabilities often involves identifying the relevant outcomes in your sample space, then dividing by the total number of possible outcomes. For example, the probability of rolling a prime number greater than 7 with two dice is 2/36 or 1/18.
👉 When calculating probabilities from experimental data, make sure your assigned probabilities for all simple events add up to exactly 1. This is a quick way to check your work!
Combination problems involve selecting groups without regard to order. The notation nCr represents the number of ways to choose r items from n distinct items.
When drawing cards or selecting committee members, combinations help calculate probabilities. For example, when drawing 7 cards from a 52-card deck, the probability of getting all hearts is the number of ways to select 7 hearts divided by the total ways to select 7 cards 13C7/52C7.
Similarly, when forming a committee from a diverse group, we use combinations to find specific group compositions. If selecting 6 people from 12 men and 16 women, the probability of getting exactly 4 men and 2 women is calculated as (12C4 × 16C2)/28C6.
🧩 Combination problems become easier when you break them into steps first count the favorable outcomes (using combinations for each group), then divide by the total possible outcomes.
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
3
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App Store
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
River Przygoda
@riverprzygoda_jvnr
This chapter covers the fundamental concepts of probability, introducing how we calculate the likelihood of events occurring in random experiments. Understanding probability is essential for predicting outcomes in uncertain situations, from simple coin flips to complex real-world scenarios.
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Probability helps us measure how likely events are to happen. The basic formula for finding the probability of an event A is P(A) = n(A)/n(U), which compares the number of favorable outcomes to the total possible outcomes.
When working with multiple events, we need specific formulas. For complementary events (something either happens or doesn't), we use P(A) + P(A') = 1. For combined events, the formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B) accounts for potential overlap.
Mutually exclusive events can't happen at the same time, simplifying our formula to P(A ∪ B) = P(A) + P(B). When events might influence each other, we use conditional probability: P(A|B) = P(A ∩ B)/P(B). If events don't affect each other, they're independent, and P(A ∩ B) = P(A)P(B).
💡 Remember that the union symbol (∪) represents "OR" situations, while the intersection symbol (∩) represents "AND" situations when calculating probabilities.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Random experiments are unpredictable processes that don't yield the same results each time, even under identical conditions. The sample space (S) contains all possible outcomes, with each individual outcome called a simple event.
Any subset of the sample space is an event (E). Events can be simple (containing just one outcome) or compound (containing multiple outcomes). For any probability calculation to make sense, probabilities must always be between 0 and 1, and all probabilities in a sample space must add up to 1.
We can determine probability through relative frequency, which is calculated as f(E)/n, where f(E) is how many times an event occurs in n trials. This approach is particularly useful when we have experimental data rather than theoretical probabilities.
🔑 When identifying events, always start by determining the appropriate sample space. For example, a two-child family could have sample space {BB, BG, GB, GG} if birth order matters, or {0, 1, 2} if you're only counting the number of girls.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
When working with dice or similar random experiments, we can map out all possible outcomes to find probabilities. For instance, when rolling two dice, there are 36 possible outcomes, and the event "sum equals 5" includes four possibilities: (1,4), (4,1), (2,3), and (3,2).
Experimental probabilities come from observed data rather than theoretical calculations. If you flip two coins 1,000 times and HH appears 273 times, you might assign P(HH) = 0.273. From this experimental data, you can calculate compound event probabilities by adding the relevant simple event probabilities.
Finding probabilities often involves identifying the relevant outcomes in your sample space, then dividing by the total number of possible outcomes. For example, the probability of rolling a prime number greater than 7 with two dice is 2/36 or 1/18.
👉 When calculating probabilities from experimental data, make sure your assigned probabilities for all simple events add up to exactly 1. This is a quick way to check your work!
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Combination problems involve selecting groups without regard to order. The notation nCr represents the number of ways to choose r items from n distinct items.
When drawing cards or selecting committee members, combinations help calculate probabilities. For example, when drawing 7 cards from a 52-card deck, the probability of getting all hearts is the number of ways to select 7 hearts divided by the total ways to select 7 cards: 13C7/52C7.
Similarly, when forming a committee from a diverse group, we use combinations to find specific group compositions. If selecting 6 people from 12 men and 16 women, the probability of getting exactly 4 men and 2 women is calculated as (12C4 × 16C2)/28C6.
🧩 Combination problems become easier when you break them into steps: first count the favorable outcomes (using combinations for each group), then divide by the total possible outcomes.
Our AI companion is specifically built for the needs of students. Based on the millions of content pieces we have on the platform we can provide truly meaningful and relevant answers to students. But its not only about answers, the companion is even more about guiding students through their daily learning challenges, with personalised study plans, quizzes or content pieces in the chat and 100% personalisation based on the students skills and developments.
You can download the app in the Google Play Store and in the Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
3
Smart Tools NEW
Transform this note into: ✓ 50+ Practice Questions ✓ Interactive Flashcards ✓ Full Mock Exam ✓ Essay Outlines
App Store
Google Play
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
The app is very easy to use and well designed. I have found everything I was looking for so far and have been able to learn a lot from the presentations! I will definitely use the app for a class assignment! And of course it also helps a lot as an inspiration.
Stefan S
iOS user
This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.
Samantha Klich
Android user
Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.
Anna
iOS user
I think it’s very much worth it and you’ll end up using it a lot once you get the hang of it and even after looking at others notes you can still ask your Artificial intelligence buddy the question and ask to simplify it if you still don’t get it!!! In the end I think it’s worth it 😊👍 ⚠️Also DID I MENTION ITS FREEE YOU DON’T HAVE TO PAY FOR ANYTHING AND STILL GET YOUR GRADES IN PERFECTLY❗️❗️⚠️
Thomas R
iOS user
Knowunity is the BEST app I’ve used in a minute. This is not an ai review or anything this is genuinely coming from a 7th grade student (I know 2011 im young) but dude this app is a 10/10 i have maintained a 3.8 gpa and have plenty of time for gaming. I love it and my mom is just happy I got good grades
Brad T
Android user
Not only did it help me find the answer but it also showed me alternative ways to solve it. I was horrible in math and science but now I have an a in both subjects. Thanks for the help🤍🤍
David K
iOS user
The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!
Sudenaz Ocak
Android user
In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.
Greenlight Bonnie
Android user
I found this app a couple years ago and it has only gotten better since then. I really love it because it can help with written questions and photo questions. Also, it can find study guides that other people have made as well as flashcard sets and practice tests. The free version is also amazing for students who might not be able to afford it. Would 100% recommend
Aubrey
iOS user
Best app if you're in Highschool or Junior high. I have been using this app for 2 school years and it's the best, it's good if you don't have anyone to help you with school work.😋🩷🎀
Marco B
iOS user
THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮
Elisha
iOS user
This app is phenomenal down to the correct info and the various topics you can study! I greatly recommend it for people who struggle with procrastination and those who need homework help. It has been perfectly accurate for world 1 history as far as I’ve seen! Geometry too!
Paul T
iOS user
