## What is Conditional Probability?

### Basic concept

**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

- 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.

### Real-life example

### 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

### Independent events

### Dependent events

### 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

**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

### 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

### 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.

## Conditional Probability in GCSE exams

### 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

**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

### 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

### 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:**

### Example problem

**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

### 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

**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.

## Frequently asked questions

**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.

**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.