# IGCSE Higher: Probability

## Scheme of work: IGCSE Higher: Year 10: Term 3: Probability

#### Prerequisite Knowledge

1. Understanding Basic Probability
• Concept: Knowing how to calculate the probability of a single event occurring.
• Example: $\text{Probability of rolling a 3 on a fair six-sided die} = \frac{1}{6}$
2. Understanding Ratios and Fractions
• Concept: Being comfortable with working with ratios and fractions, as probability is often expressed in these terms.
• Example: $\text{Probability of drawing a red ball from a bag with 2 red and 3 blue balls} = \frac{2}{5}$

#### Success Criteria

1. Drawing and Using Tree Diagrams
• Performing the Task: Students should be able to draw tree diagrams to represent multiple stages of a probability scenario.
• Example: $\text{Draw a tree diagram for flipping two coins}$
• Describing the Diagram: Students should be able to use tree diagrams to calculate the probabilities of combined events.
• Example: $\text{Calculate the probability of getting at least one head when flipping two coins using a tree diagram}$
2. Determining the Probability of Independent Events
• Performing the Calculation: Students should be able to calculate the probability that two or more independent events will occur.
• Example: $\text{Probability of rolling a 3 on a die and flipping a head on a coin} = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$
• Describing Independent Events: Students should be able to describe the concept of independent events in probability.
• Example: $\text{Rolling a die and flipping a coin are independent events because the outcome of one does not affect the outcome of the other}$
3. Using Conditional Probability
• Performing the Calculation: Students should be able to use conditional probability to find the probability of combined events.
• Example: $\text{Probability of picking two red balls from a bag without replacement} = \frac{2}{5} \times \frac{1}{4} = \frac{2}{20} = \frac{1}{10}$
• Describing Conditional Probability: Students should be able to describe how conditional probability works when combining events.
• Example: $\text{The probability of the second event depends on the outcome of the first event when picking two balls without replacement}$
4. Applying Probability to Simple Problems
• Performing the Calculation: Students should be able to apply probability to solve simple real-world problems.
• Example: $\text{Probability of drawing an ace from a standard deck of cards} = \frac{4}{52} = \frac{1}{13}$
• Describing the Application: Students should be able to describe how probability is applied in different scenarios.
• Example: $\text{Probability is used to predict outcomes in games of chance, such as cards and dice}$

#### Key Concepts

1. Drawing and Using Tree Diagrams
• Performing the Task: Students should be able to draw tree diagrams to represent multiple stages of a probability scenario.
• Example: $\text{Draw a tree diagram for flipping two coins}$
• Describing the Diagram: Students should be able to use tree diagrams to calculate the probabilities of combined events.
• Example: $\text{Calculate the probability of getting at least one head when flipping two coins using a tree diagram}$
2. Determining the Probability of Independent Events
• Performing the Calculation: Students should be able to calculate the probability that two or more independent events will occur.
• Example: $\text{Probability of rolling a 3 on a die and flipping a head on a coin} = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$
• Describing Independent Events: Students should be able to describe the concept of independent events in probability.
• Example: $\text{Rolling a die and flipping a coin are independent events because the outcome of one does not affect the outcome of the other}$
3. Using Conditional Probability
• Performing the Calculation: Students should be able to use conditional probability to find the probability of combined events.
• Example: $\text{Probability of picking two red balls from a bag without replacement} = \frac{2}{5} \times \frac{1}{4} = \frac{2}{20} = \frac{1}{10}$
• Describing Conditional Probability: Students should be able to describe how conditional probability works when combining events.
• Example: $\text{The probability of the second event depends on the outcome of the first event when picking two balls without replacement}$
4. Applying Probability to Simple Problems
• Performing the Calculation: Students should be able to apply probability to solve simple real-world problems.
• Example: $\text{Probability of drawing an ace from a standard deck of cards} = \frac{4}{52} = \frac{1}{13}$
• Describing the Application: Students should be able to describe how probability is applied in different scenarios.
• Example: $\text{Probability is used to predict outcomes in games of chance, such as cards and dice}$

#### Common Misconceptions

1. Tree Diagrams
• Common Mistake: Students might incorrectly draw tree diagrams, missing some branches or incorrectly labeling probabilities.
• Example: Missing a branch for one of the outcomes when flipping two coins, such as forgetting the (Tails, Tails) outcome.
• Common Mistake: Students might incorrectly use tree diagrams to calculate probabilities, leading to incorrect results.
• Example: Incorrectly multiplying probabilities along the branches of a tree diagram.
2. Independent Events
• Common Mistake: Students might incorrectly treat dependent events as independent, leading to incorrect calculations.
• Example: Calculating the probability of drawing two aces in a row from a deck of cards with replacement as independent events: $\text{Incorrect: } \left( \frac{4}{52} \right)^2 = \frac{1}{169}$ $\text{Correct: } \frac{4}{52} \times \frac{3}{51} = \frac{1}{221}$
• Common Mistake: Students might incorrectly assume that the probability of combined independent events is simply the sum of individual probabilities rather than the product.
• Example: Calculating the probability of rolling a 3 on a die and flipping a head on a coin as: $\text{Incorrect: } \frac{1}{6} + \frac{1}{2} = \frac{2}{3}$ $\text{Correct: } \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}$
3. Conditional Probability
• Common Mistake: Students might incorrectly calculate conditional probabilities by not adjusting the sample space after the first event.
• Example: Calculating the probability of picking two red balls from a bag without replacement as: $\text{Incorrect: } \frac{2}{5} \times \frac{2}{5} = \frac{4}{25}$ $\text{Correct: } \frac{2}{5} \times \frac{1}{4} = \frac{1}{10}$
• Common Mistake: Students might confuse the order of events when calculating conditional probabilities.
• Example: Reversing the order of events in conditional probability calculations, such as treating the probability of drawing a red ball first and then a blue ball as the same as drawing a blue ball first and then a red ball.
4. Applying Probability
• Common Mistake: Students might incorrectly apply probability concepts to real-world problems by not clearly defining the events.
• Example: Misinterpreting the problem scenario and calculating the probability of drawing an ace from a standard deck of cards as: $\text{Incorrect: } \frac{4}{52} \times \frac{4}{52} = \frac{16}{2704}$ $\text{Correct: } \frac{4}{52} = \frac{1}{13}$
• Common Mistake: Students might assume probabilities add instead of multiply in multi-step problems.
• Example: Calculating the probability of getting heads in two consecutive coin flips as: $\text{Incorrect: } \frac{1}{2} + \frac{1}{2} = 1$ $\text{Correct: } \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

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