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Understanding Probability Basics
Think of probability as your mathematical crystal ball - it helps you predict how likely different outcomes are when you conduct an experiment or trial. Whether you're flipping coins or analysing data, probability gives you a numerical way to express likelihood.
There are two main ways to estimate probability. Experimental probability uses real data you've collected - like throwing a drawing pin 100 times and seeing how often it lands point-up. The formula is P(U) = n(U)/n(T), where n(U) is the number of successful outcomes and n(T) is your total trials.
Theoretical probability works differently - you use logic rather than experiments. For a fair coin toss, you know there's a 1 in 2 chance of getting heads, so P(H) = 1/2. This approach works brilliantly when all outcomes are equally likely.
Key insight: Probability always sits between 0 and 1. A certain event has probability 1, while an impossible event has probability 0.
Working with Multiple Events
Real life rarely involves single, simple events - you'll often need to calculate probabilities when multiple things could happen. This is where understanding complements and unions becomes essential for your toolkit.
The complement of an event A (written as A') represents everything that isn't A. Since something must either happen or not happen, P(A) + P(A') = 1. This relationship is incredibly useful for solving complex problems by working backwards.
When dealing with multiple events, you'll encounter mutually exclusive events - these cannot happen simultaneously (like rolling a 3 and a 5 on the same die). For these events, P(A ∪ B) = P(A) + P(B). However, if events aren't mutually exclusive, you must subtract the overlap: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
Watch out: The union symbol (∪) means "or" whilst the intersection symbol (∩) means "and" - getting these mixed up is a common mistake that can cost you marks.
Advanced Probability Concepts
Understanding sample spaces (all possible outcomes) and independent events will elevate your probability skills to the next level. Independent events are particularly important - the outcome of one doesn't affect the other, like consecutive coin flips.
For independent events, multiply probabilities: P(A ∩ B) = P(A) × P(B). This multiplication rule is crucial for tree diagrams, where you multiply along branches and add up different pathways to find total probabilities.
Expected frequency helps you predict real-world outcomes. If an event has probability 3/4 and you conduct 400 trials, you'd expect it to occur 400 × 3/4 = 300 times. This connects theoretical probability with practical applications.
Pro tip: Tree diagrams are your best friend for complex probability problems - they help visualise all possible outcomes and make calculations much clearer.
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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.
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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
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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
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Best app on earth! no words because it’s too good
Thomas R
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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 in A Level Maths
Probability is essentially a mathematical way to measure how likely something is to happen - from predicting coin tosses to estimating real-world events. You'll learn two main approaches: experimental (using actual data) and theoretical (using mathematical reasoning), plus crucial concepts... Show more
Understanding Probability Basics
Think of probability as your mathematical crystal ball - it helps you predict how likely different outcomes are when you conduct an experiment or trial. Whether you're flipping coins or analysing data, probability gives you a numerical way to express likelihood.
There are two main ways to estimate probability. Experimental probability uses real data you've collected - like throwing a drawing pin 100 times and seeing how often it lands point-up. The formula is P(U) = n(U)/n(T), where n(U) is the number of successful outcomes and n(T) is your total trials.
Theoretical probability works differently - you use logic rather than experiments. For a fair coin toss, you know there's a 1 in 2 chance of getting heads, so P(H) = 1/2. This approach works brilliantly when all outcomes are equally likely.
Key insight: Probability always sits between 0 and 1. A certain event has probability 1, while an impossible event has probability 0.
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Working with Multiple Events
Real life rarely involves single, simple events - you'll often need to calculate probabilities when multiple things could happen. This is where understanding complements and unions becomes essential for your toolkit.
The complement of an event A (written as A') represents everything that isn't A. Since something must either happen or not happen, P(A) + P(A') = 1. This relationship is incredibly useful for solving complex problems by working backwards.
When dealing with multiple events, you'll encounter mutually exclusive events - these cannot happen simultaneously (like rolling a 3 and a 5 on the same die). For these events, P(A ∪ B) = P(A) + P(B). However, if events aren't mutually exclusive, you must subtract the overlap: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
Watch out: The union symbol (∪) means "or" whilst the intersection symbol (∩) means "and" - getting these mixed up is a common mistake that can cost you marks.
Sign up to see the contentIt's free!
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Advanced Probability Concepts
Understanding sample spaces (all possible outcomes) and independent events will elevate your probability skills to the next level. Independent events are particularly important - the outcome of one doesn't affect the other, like consecutive coin flips.
For independent events, multiply probabilities: P(A ∩ B) = P(A) × P(B). This multiplication rule is crucial for tree diagrams, where you multiply along branches and add up different pathways to find total probabilities.
Expected frequency helps you predict real-world outcomes. If an event has probability 3/4 and you conduct 400 trials, you'd expect it to occur 400 × 3/4 = 300 times. This connects theoretical probability with practical applications.
Pro tip: Tree diagrams are your best friend for complex probability problems - they help visualise all possible outcomes and make calculations much clearer.
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What is the Knowunity AI companion?
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Students love us — and so will you.
4.9/5
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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