Probability is all about working out how likely something is... Show more
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24 Dec 2025
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Understanding Probability
I
Itz Gamer Al 123
@itzgameral123_tyei
1 / 4
Probability and the Probability Scale
Ever wondered why weather forecasts talk about "30% chance of rain"? That's probability in action - a way of measuring how likely something is to happen.
The probability scale runs from 0 to 1, where 0 means impossible and 1 means certain. You can write probabilities as fractions (like ½), decimals (like 0.5), or percentages (like 50%). Think of it like a sliding scale: picking a blue counter from a bag of only red counters sits at 0, whilst Monday following Sunday sits firmly at 1.
To calculate theoretical probability, use this simple formula: number of ways the outcome can happen ÷ total number of possible outcomes. For example, rolling a 4 on a dice gives you 1 ÷ 6 = ⅙, because there's one way to get a 4 out of six possible numbers.
Quick Check: Rolling an odd number on a dice? There are three odd numbers (1, 3, 5) out of six possible outcomes, so that's 3 ÷ 6 = ½!
Relative Frequency and Experimental Probability
Real life isn't always as neat as theory suggests. Relative frequency (also called experimental probability) shows what actually happens when you run experiments in the real world.
If you flip a coin 100 times, theory says you should get 50 heads. But you might actually get 59 heads and 41 tails, making your relative frequency 59/100 or 59%. This approach becomes essential when dealing with biased dice or unfair coins, where theoretical probability doesn't apply.
The more times you repeat an experiment, the more reliable your relative frequency becomes. It's like getting a clearer picture by adding more data points - 50 coin flips gives you a better estimate than just 10 flips.
Real-World Tip: Sports statistics use relative frequency all the time - a footballer's goal-scoring record is based on past performance, not theoretical calculations!
Combined Events and Systematic Approaches
When multiple events happen together, you need organised methods to avoid missing possibilities. Systematic listing helps you find all outcomes in a logical order.
Frequency trees organise information visually when you're given actual numbers rather than probabilities. For example, if a running club has 160 members with specific breakdowns by gender and age, you can map this out systematically to calculate probabilities for randomly selected members.
Tree diagrams show all possible outcomes when events happen in sequence. Each branch represents a possible outcome with its probability clearly marked. For independent events (like rolling dice twice), the outcome of the first event doesn't affect the second.
Memory Trick: With tree diagrams, multiply probabilities along branches to find the probability of that specific path happening.
Working with Tree Diagrams
Tree diagrams become powerful tools when you need to calculate probabilities for sequences of events. Each path through the tree represents one possible outcome.
To find the probability of any specific combination, simply multiply the probabilities along that branch. For instance, rolling two sixes in a row: ⅙ × ⅙ = 1/36.
The beauty of tree diagrams lies in their completeness - they show every possible outcome clearly. Whether you're looking at rolling dice, drawing cards, or any other sequential events, the visual layout helps you avoid missing crucial combinations.
Pro Tip: Always check that all your final probabilities add up to 1 - this confirms you haven't missed any outcomes or made calculation errors!
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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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user
Understanding Probability
I
Itz Gamer Al 123
@itzgameral123_tyei
Probability is all about working out how likely something is to happen - from impossible (0) to certain (1). Whether you're flipping coins, rolling dice, or predicting real-world events, understanding probability helps you make sense of chance and uncertainty in... Show more
Probability and the Probability Scale
Ever wondered why weather forecasts talk about "30% chance of rain"? That's probability in action - a way of measuring how likely something is to happen.
The probability scale runs from 0 to 1, where 0 means impossible and 1 means certain. You can write probabilities as fractions (like ½), decimals (like 0.5), or percentages (like 50%). Think of it like a sliding scale: picking a blue counter from a bag of only red counters sits at 0, whilst Monday following Sunday sits firmly at 1.
To calculate theoretical probability, use this simple formula: number of ways the outcome can happen ÷ total number of possible outcomes. For example, rolling a 4 on a dice gives you 1 ÷ 6 = ⅙, because there's one way to get a 4 out of six possible numbers.
Quick Check: Rolling an odd number on a dice? There are three odd numbers (1, 3, 5) out of six possible outcomes, so that's 3 ÷ 6 = ½!
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Relative Frequency and Experimental Probability
Real life isn't always as neat as theory suggests. Relative frequency (also called experimental probability) shows what actually happens when you run experiments in the real world.
If you flip a coin 100 times, theory says you should get 50 heads. But you might actually get 59 heads and 41 tails, making your relative frequency 59/100 or 59%. This approach becomes essential when dealing with biased dice or unfair coins, where theoretical probability doesn't apply.
The more times you repeat an experiment, the more reliable your relative frequency becomes. It's like getting a clearer picture by adding more data points - 50 coin flips gives you a better estimate than just 10 flips.
Real-World Tip: Sports statistics use relative frequency all the time - a footballer's goal-scoring record is based on past performance, not theoretical calculations!
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Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Combined Events and Systematic Approaches
When multiple events happen together, you need organised methods to avoid missing possibilities. Systematic listing helps you find all outcomes in a logical order.
Frequency trees organise information visually when you're given actual numbers rather than probabilities. For example, if a running club has 160 members with specific breakdowns by gender and age, you can map this out systematically to calculate probabilities for randomly selected members.
Tree diagrams show all possible outcomes when events happen in sequence. Each branch represents a possible outcome with its probability clearly marked. For independent events (like rolling dice twice), the outcome of the first event doesn't affect the second.
Memory Trick: With tree diagrams, multiply probabilities along branches to find the probability of that specific path happening.
Sign up to see the contentIt's free!
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Improve your grades
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Working with Tree Diagrams
Tree diagrams become powerful tools when you need to calculate probabilities for sequences of events. Each path through the tree represents one possible outcome.
To find the probability of any specific combination, simply multiply the probabilities along that branch. For instance, rolling two sixes in a row: ⅙ × ⅙ = 1/36.
The beauty of tree diagrams lies in their completeness - they show every possible outcome clearly. Whether you're looking at rolling dice, drawing cards, or any other sequential events, the visual layout helps you avoid missing crucial combinations.
Pro Tip: Always check that all your final probabilities add up to 1 - this confirms you haven't missed any outcomes or made calculation errors!
We thought you’d never ask...
What is the Knowunity AI companion?
Our AI Companion is a student-focused AI tool that offers more than just answers. Built on millions of Knowunity resources, it provides relevant information, personalised study plans, quizzes, and content directly in the chat, adapting to your individual learning journey.
Where can I download the Knowunity app?
You can download the app from Google Play Store and Apple App Store.
Is Knowunity really free of charge?
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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Can't find what you're looking for? Explore other subjects.
Students love us — and so will you.
4.9/5
App Store
4.8/5
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
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
Best app on earth! no words because it’s too good
Thomas R
iOS user
Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.
Basil
Android user
This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.
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
very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.
Rohan U
Android user
I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.
Xander S
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 apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now
Paul T
iOS user