Conditional probability tells us how likely an event will happen based on another event that has taken place. It's an important topic in probability theory and a challenging area of the GCSE Maths curriculum. This article will explain conditional probability and how to calculate it. Examples and quiz questions have been included to test your knowledge.
If you need further help, TeachTutti has a list of qualified GCSE Maths tutors who can explain this topic in the GCSE curriculum.
What is Conditional Probability?
Conditional probability looks at how likely an event will happen after another event has taken place. It helps us to understand in probability theory how the likelihood of an event happening changes when we have more information.
Basic concept
Let's run through an example to explore the basic concept. We have a deck of 52 playing cards. A player draws a king from the deck and they don't replace it. How likely is it that the next card drawn is an ace?
- Find all the possible outcomes: 51 cards are remaining after drawing the king.
- Find the number of desired outcomes: There are 4 aces in a full deck. We have drawn a kind, so there are 4 aces left in the remaining 51 cards.
- Calculate the conditional probability: The probability of drawing an ace given that a king has already been drawn is: P(Ace | King) = 4 / 51.
Mathematical Definition
This is the conditional probability formula: P(A|B) = P(A and B) / P(B)
- P(A|B) is the conditional probability of event A occurring when event B has occurred.
- P(A and B) is the joint probability of both events A and B happening.
- P(B) is the probability of event B occurring.
This formula shows that we need to know the joint probability of both events happening and the probability of the conditioning event to find the conditional probability.
Real-life example
You're conducting a school survey to find out how likely a student will like both maths and science. 30% like maths, 25% like science and 15% of students like both subjects. What is the probability that a random student you pick will like both subjects?
P(Maths | Science) = P(Maths and Science) / P(Science) = 0.15 / 0.25 = 0.6.
The formula above tells us that there's a 60% chance a student will like maths if they also like science.
1
There is a bag with 3 red balls, 5 blue balls and 2 green balls. A ball that isn't blue is drawn from the bag. What is the probability that the next ball will be red?
Key concepts and terms
There are several important terms you need to understand to tackle conditional probability.
Independent events
The outcomes of Independent events don't affect each other. Rolling a die is a good example: it is independent because the outcome of rolling the die won't affect the next time you roll it.
The formula for independent events shows the probability of both events happening is simply the product of their probabilities:
P(A and B) = P(A) × P(B).
Dependent events
The outcome of dependent events will affect the other event. These are the events that conditional probability mainly deals with. For example, if you remove a card from a deck and don't replace it, the next draw is affected by the reduced deck.
The formula below finds the conditional probability of event A happening when event B has taken place:
P(A|B) = P(A and B) / P(B).
Notation and symbols
- P(A|B): Probability of event A happening when event B has occurred.
- P(A and B): Probability of both events A and B taking place.
- P(AUB): Probability of either event A or B occurring (union).
- P(ANB): Probability of both events A and B taking place (intersection).
Venn diagrams and Tree diagrams
Venn diagrams: This diagram shows sets and the relationships between them. They can show the probability of combined events and conditional probabilities:
Tree diagrams: These diagrams help us to visualise sequential events. Each branch shows all the possible outcomes. The probabilities in a branch help us to calculate conditional probabilities:
Example: Using a tree diagram
You have a bag with 3 red balls and 2 blue balls. You draw out a ball, make a note of its colour and draw another ball without replacing the first.
1. Draw the tree diagram:
- First draw: 3 red (R), 2 blue (B)
- Second draw if the first ball is red: 2R, 2B
- Second draw if the first ball is blue: 3R, 1B
2. Calculate the probabilities:
- The probability of drawing a red ball first: 3/5
- The probability of drawing a red ball second if the first was red: 2/4 = 1/2
- The probability of drawing a red ball second if the first was blue: 3/4
We can use the probabilities above to calculate the conditional probability for various outcomes.
2
There is a deck of 52 cards. A spade is drawn from the deck - it isn't replaced. What's the probability that the next card is also a spade?
Calculating Conditional Probability
There are systematic steps to calculating conditional probability. They will help you tackle any problem related to the topic by breaking it down into its simplest form.
Step-by-step guide
1. Find the events:
- What events are you interested in? For a case study, let’s say event A is drawing an ace from a deck of cards and event B is drawing a king.
2. Determine the probabilities:
- Calculate the probability of each event happening.
- P(B) is the probability of drawing a king. There are 4 kings in a normal deck of 52 cards. P(B) = 4/52 = 1/13P(B)
- Find the joint probability of both events happening (if needed).
- P(A and B) is the likelihood of drawing an ace and then a king. There are 4 aces and 4 kings: P(A and B) = 4/52 × 4/51 = 4/52 × 1/51 = 1/663
3. Use the formula:
- Use the formula P(A|B) = P(A and B) / P(B).
- Add the values to the above formula: P(A|B) = P(A and B) / P(B) = 1/663 / 1/13 = 1/663 × 13 = 13/663 = 1/51.
In our case study, the conditional probability of getting an ace when a king has already been drawn is 1/51.
Conditional Probability in GCSE exams
Conditional probability commonly features in GCSE Maths. There is a range of questions on the topic, from straightforward calculations to complex problems with multiple steps and different scenarios.
Typical questions
1. Simple Conditional Probability questions:
- These questions normally ask the probability of an event when another event has happened:
- "A bag has 4 red, 3 blue and 2 green balls. One red ball is drawn. What is the probability the next ball drawn is blue?"
2. Two-Way tables:
- Two-way tables show the frequency of events and their intersections. A question using this table may ask you to find the conditional probability:
- "100 A Level students are surveyed for their chosen subject. 40 study Maths, 30 study Science and 20 study both. What is the probability that a student will study Science when they also study Maths?"
3. Tree Diagrams:
- Tree diagrams show all the possible outcomes of sequences of events. A common question with tree diagrams is to calculate probabilities along different branches:
- "A coin is tossed twice. What is the probability of getting heads after the first toss was heads?"
How to answer these questions
1. Scrutinise the question:
- Find the given condition and the event you need the probability for.
- Write down clearly what you know and what you need to find.
2. Use the formula:
- The formula is P(A|B) = P(A and B) / P(B).
- Make sure you can find P(A and B) and P(B) from the information in the problem.
3. Draw diagrams:
- A two-way table or a tree diagram may be useful for complex problems. They help visualise the problem so you can organise the information rather than keeping it in your head, thus avoiding potential problems.
Example problem
There are 5 white balls, 3 black balls and 2 red balls in a box. Two balls are drawn without replacement. What is the probability that the second ball is red when the first ball is white?
1. Find the events:
- Event A: Drawing a red ball.
- Event B: Drawing a white ball first.
2. Determine the Probabilities:
- The probability of drawing a white ball first. P(B): P(B) = 5/10 = 1/2
- The probability of drawing a red ball second given the first was white. P(A|B): After drawing a white ball, there are 9 balls left (5 white, 3 black, 2 red).
3. Calculate the Conditional Probability:
- Probability of drawing a red ball from the remaining balls: P(A|B) = 2/9
The probability of drawing a red ball when the first ball was white is 2/9.
3
There are 15 students in a school classroom: 9 girls and 6 boys. A random student is chosen and she is a girl. What is the probability that the next random student will be a boy?
Common mistakes
Conditional probability can be complex and there are common mistakes to watch out for. We have covered these below to help you in your calculations.
Common mistakes and tips
1. Forget to update the total outcomes:
- You need to update the total possible outcomes when an event has happened. For example, if you draw a card from a deck and replace it, the total remaining cards and possible outcomes decrease by one.
2. Confuse independent and dependent events:
- Conditional probability normally deals with dependent events where an event is affected by the outcome of the other event. Ensure you identify the correct type of event when doing your calculations.
3. Incorrectly use the formula:
- Make sure you use the formula correctly, both initially and when you apply values to it. Incorrect identification will lead to wrong answers.
4. Overlook conditions:
- Make sure you don't overlook conditions that are expressed in the problem. Pay close attention to the detail in the question and adjust your probabilities accordingly.
Tips for success
1. Practice:
- Regular practice will lead to proficiency in conditional probability. Go through a variety of problems so you're comfortable with different scenarios and types of questions.
2. Visual aids:
- Venn diagrams and tree diagrams help visualise the problem. Tree diagrams in particular will show all the potential probabilities, organising and displaying the information so you can more easily see the relationship between events.
3. Break down problems:
- With complex and simple problems alike, it is a good approach to break them down into smaller steps. This mechanical approach will help to avoid confusion and errors.
4. Double-check your work:
Go back and check your calculations. Make sure you have considered all the details in the problem and applied the formula correctly in your calculations.
Example problem
120 students out of 200 like to eat chocolate. 50 students like vanilla and 50 students like both. How likely is it that a student will like both chocolate and vanilla?
1. Find the events:
- Event A: Liking chocolate.
- Event B: Liking vanilla.
2. Determine the probabilities:
- Probability of liking vanilla, P(B): P(B) = 80/200 = 2/5
- Joint probability of liking both chocolate and vanilla, P(A and B): P(A and B) = 50/200 = 1/4
3. Calculate the Conditional Probability:
- P(A|B) = 1/4 / 2/5 = 1/4 × 5/2 = 5/8
The probability that a student who likes vanilla will also like chocolate is 5/8.
4
A company makes a product in two factories. 60% is created in factory A and 40% in factory B. 70% of the products in factory A pass quality control compared to 90% in factory B. What is the probability that a product that passed quality control was made in factory A?
Conclusion
Conditional probability lets you calculate how likely an event will happen based on another event occurring. It's useful for problem-solving and has applications in your exams and real-world scenarios:
- Definition: It measures how likely an event will happen when another event has taken place. It differs from simple probability, which doesn't consider prior conditions.
- Key concepts: We looked at independent and dependent events and how to notate and use symbols properly in conditional probability calculations.
- Calculating conditional probability: We explored how to calculate conditional probability with real-life examples and the formula P(A|B) = P(A and B) / P(B)
- Common mistakes: Be aware of common pitfalls to avoid errors in your probability calculations.
If you want a rounded understanding of probability, TeachTutti has written an article explaining the wider topic of probability for GCSE Maths. It's also a good idea to learn about Gerolamo Cardano who was one of the founders of probability in the 16th century.
If you're still struggling with conditional probability, TeachTutti has a list of verified GCSE Maths tutors to provide personalised learning and who can support your GCSE Maths revision.
Frequently asked questions
Conditional probability is the likelihood of an event happening when another has already occurred.
The formula for calculating conditional probability is P(A|B) = P(A and B) / P(B). P(A|B) is how likely event A will happen after event B has occurred. P(A and B) is how likely both events will happen. P(B) is the probability of event B.
Independent events: The outcome of one event will not affect the other event. A common example is flipping a coin: flipping heads first doesn't mean the second coin flip will return tails.
Dependent events: The outcome of one event affects the other event. For example, if you draw a card from a deck and don't replace it, it reduces the likelihood that you will draw another card from the same set.
- Forgetting to update the total number of outcomes after an event has taken place.
- Confusing independent and dependent events.
- Incorrectly using the conditional probability formula.
- Ignoring given conditions in the problem.
- Practice: It's irritating to say but practice does make perfect. Try to tackle a variety of problems to understand different scenarios.
- Use visual aids: Illustration including Venn and tree diagrams can help you visualise problems.
- Break down problems: Use a methodical, step-by-step approach to break complex problems down into their smallest, simplest form.
Glossary
- Conditional probability - The probability an event will happen when another event has taken place.
- Independent events - An event that doesn't affect the outcome of the other event.
- Dependent events - An event that affects the outcome of the other event.
- Joint probability - The likelihood of two events happening at the same time.
- Two-way table - A table showing the frequency of different combinations of two categorical variables.
- Tree diagram - A diagram that shows the possible outcomes of a sequence of events as probabilities.
- Venn diagram - A diagram displaying all the possible logical relations between a finite collection of different sets. This is often used to show probabilities.
- Probability formula - A mathematical formula for calculating the probability of an event.
- Sample space - The set of all possible outcomes in a probability experiment.
- Event - A specific outcome/set from an experiment.
- Union - The probability of either event A or event B occurring.
- Intersection - The probability of both events A and B occurring.
