Maths
Chemistry
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English Language
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Responding to change (a2 only)
Infection and response
Homeostasis and response
Energy transfers (a2 only)
Cell biology
Organisms respond to changes in their internal and external environments (a-level only)
Biological molecules
Organisation
Substance exchange
Bioenergetics
Genetic information & variation
Inheritance, variation and evolution
Genetics & ecosystems (a2 only)
Ecology
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Britain & the wider world: 1745 -1901
1l the quest for political stability: germany, 1871-1991
The cold war
Inter-war germany
Medieval period: 1066 -1509
2d religious conflict and the church in england, c1529-c1570
2o democracy and nazism: germany, 1918-1945
1f industrialisation and the people: britain, c1783-1885
1c the tudors: england, 1485-1603
2m wars and welfare: britain in transition, 1906-1957
World war two & the holocaust
2n revolution and dictatorship: russia, 1917-1953
2s the making of modern britain, 1951-2007
World war one
Britain: 1509 -1745
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14
0
Jessie M
05/12/2025
Maths
Probability - A Level
607
•
5 Dec 2025
•
Probability is everywhere - from predicting weather to calculating your... Show more
Ever wondered how to calculate the odds of rolling a specific number on a dice? Probability gives you the tools to work this out mathematically. When outcomes have equal chances of happening, we call them equiprobable.
Sample space diagrams are brilliant for visualising all possible outcomes. Think of flipping two coins - you can map out every combination . The basic probability formula is P(A) = n(A)/n(ξ), where n(A) is the number of ways event A can occur, and n(ξ) is the total number of possible outcomes.
The complement of an event (written as A') represents everything that isn't A. If there's a 0.3 chance of rain, there's a 0.7 chance it won't rain. This gives us P(A') = 1 - P(A).
Quick Tip: Venn diagrams help visualise probability problems. The union (A ∪ B) means "A or B happening", whilst the intersection (A ∩ B) means "both A and B happening".
Real-life probability gets more interesting when events influence each other. Conditional probability asks: "What's the chance of B happening if we know A has already occurred?" It's written as P(B|A).
The formula P(B|A) = P(B ∩ A)/P(A) might look scary, but it's quite logical. You're essentially narrowing down your focus to just the cases where A happened. Think about age affecting health risks - knowing someone's age changes your estimate of their health probability.
Independent events don't affect each other (like rolling dice twice), so P(B|A) = P(B). Dependent events do influence each other, making P(B|A) ≠ P(B|A'). This distinction is crucial for exam questions.
Remember: For dependent events, P(A ∩ B) = P(A) × P(B|A). For independent events, it's simply P(A) × P(B).
Two-way tables transform messy probability problems into manageable calculations. In the employee training example, you've got 256 total employees split between trained/untrained and employed/left.
Finding basic probabilities is straightforward - just divide the relevant number by the total (256). For P(T), you get 152/256 because 152 employees received training. Intersection probabilities like P(T ∩ S) use the overlap - 109 employees who were both trained and stayed.
Conditional probability becomes much clearer with tables. P(S|T) asks: "Of those who received training, what fraction stayed?" You divide 109 (trained and stayed) by 152 (total trained) to get your answer.
Exam Strategy: Always check for the total symbol (ξ) in exam questions - it's your denominator for basic probability calculations.
Tree diagrams are your best friend for multi-stage probability problems. They map out all possible paths through dependent events, making calculations much clearer than trying to work everything out in your head.
Each branch shows a probability, and you multiply along paths to find specific outcomes. For dependent events, the second set of branches changes based on what happened first - that's where conditional probability comes in.
The key insight is that P(A ∩ B) = P(A) × P(B|A) for any two dependent events. This formula works whether you're dealing with medical diagnoses, quality control, or any scenario where the first event affects the second.
Pro Tip: Tree diagrams help you spot all possible combinations and ensure your probabilities add up to 1 - a great way to check your working.
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.
You can download the app from Google Play Store and Apple App Store.
That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.
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
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
Probability is everywhere - from predicting weather to calculating your chances of passing an exam. This chapter covers the essential probability concepts you'll need for your exams, including theoretical calculations, conditional probability, and how to work with dependent and independent... Show more
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Improve your grades
Join milions of students
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Ever wondered how to calculate the odds of rolling a specific number on a dice? Probability gives you the tools to work this out mathematically. When outcomes have equal chances of happening, we call them equiprobable.
Sample space diagrams are brilliant for visualising all possible outcomes. Think of flipping two coins - you can map out every combination . The basic probability formula is P(A) = n(A)/n(ξ), where n(A) is the number of ways event A can occur, and n(ξ) is the total number of possible outcomes.
The complement of an event (written as A') represents everything that isn't A. If there's a 0.3 chance of rain, there's a 0.7 chance it won't rain. This gives us P(A') = 1 - P(A).
Quick Tip: Venn diagrams help visualise probability problems. The union (A ∪ B) means "A or B happening", whilst the intersection (A ∩ B) means "both A and B happening".
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Improve your grades
Join milions of students
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Real-life probability gets more interesting when events influence each other. Conditional probability asks: "What's the chance of B happening if we know A has already occurred?" It's written as P(B|A).
The formula P(B|A) = P(B ∩ A)/P(A) might look scary, but it's quite logical. You're essentially narrowing down your focus to just the cases where A happened. Think about age affecting health risks - knowing someone's age changes your estimate of their health probability.
Independent events don't affect each other (like rolling dice twice), so P(B|A) = P(B). Dependent events do influence each other, making P(B|A) ≠ P(B|A'). This distinction is crucial for exam questions.
Remember: For dependent events, P(A ∩ B) = P(A) × P(B|A). For independent events, it's simply P(A) × P(B).
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Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Two-way tables transform messy probability problems into manageable calculations. In the employee training example, you've got 256 total employees split between trained/untrained and employed/left.
Finding basic probabilities is straightforward - just divide the relevant number by the total (256). For P(T), you get 152/256 because 152 employees received training. Intersection probabilities like P(T ∩ S) use the overlap - 109 employees who were both trained and stayed.
Conditional probability becomes much clearer with tables. P(S|T) asks: "Of those who received training, what fraction stayed?" You divide 109 (trained and stayed) by 152 (total trained) to get your answer.
Exam Strategy: Always check for the total symbol (ξ) in exam questions - it's your denominator for basic probability calculations.
Access to all documents
Improve your grades
Join milions of students
By signing up you accept Terms of Service and Privacy Policy
Tree diagrams are your best friend for multi-stage probability problems. They map out all possible paths through dependent events, making calculations much clearer than trying to work everything out in your head.
Each branch shows a probability, and you multiply along paths to find specific outcomes. For dependent events, the second set of branches changes based on what happened first - that's where conditional probability comes in.
The key insight is that P(A ∩ B) = P(A) × P(B|A) for any two dependent events. This formula works whether you're dealing with medical diagnoses, quality control, or any scenario where the first event affects the second.
Pro Tip: Tree diagrams help you spot all possible combinations and ensure your probabilities add up to 1 - a great way to check your working.
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.
You can download the app from Google Play Store and Apple App Store.
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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Explore key probability concepts including intersection, union, mutually exclusive events, and conditional probability. This summary provides clear explanations and examples, ideal for GCSE Maths students preparing for exams. Understand the Addition Rule and how to apply Venn diagrams and Karnaugh maps effectively.
MathsExplore essential concepts in probability, including intersection, union, and mutually exclusive events. This summary covers key probability facts and notation, making it ideal for GCSE Maths students at both higher and foundation levels.
MathsExplore key probability concepts including independent and mutually exclusive events, the addition rule, and both theoretical and experimental probability. This summary provides essential formulas and examples to enhance your understanding of probability in A Level Maths. Ideal for AQA Statistics students.
MathsExplore essential GCSE probability concepts including experimental and theoretical probability, conditional probability, and event intersections. This summary covers key definitions and examples to enhance your understanding and exam preparation.
MathsExplore key concepts in probability, including the Addition Rule, mutually exclusive events, and calculations for expected frequency and relative frequency. This summary provides essential insights for mastering probability in statistics.
MathsEnhance your understanding of probability with this comprehensive workbook. Covering key topics such as expected value, theoretical probability, and probability models, this resource is designed for students aged 11-16. Engage with practical exercises, including spinner probabilities, Venn diagrams, and binomial probability, to solidify your numeracy skills and prepare for exams.
MathsApp 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
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
Sociology