1. Overview
Probability is the measure of how likely an event is to occur. It's a fundamental concept in mathematics that quantifies uncertainty. The probability of an event is always a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. This topic introduces the basic principles of probability, including calculating the probability of single events and understanding complementary events. These skills are crucial for solving a wide range of problems, from simple games of chance to more complex scenarios.
Key Definitions
- Experiment: A process, such as flipping a coin or rolling a die, that has a number of possible outcomes.
- Outcome: A possible result of an experiment (e.g., getting a '4' on a die).
- Sample Space: The set of all possible outcomes of an experiment.
- Event: One or more outcomes of an experiment (e.g., rolling an even number).
- Theoretical Probability: Probability based on reasoning and calculated using a formula.
- Complementary Event: The probability of an event not happening.
Core Content
3.1 The Probability Scale
Probability is always measured on a scale from 0 to 1.
- A probability of 0 means the event is impossible.
- A probability of 1 means the event is certain.
- Probabilities can be written as fractions, decimals, or percentages (0% to 100%).
3.2 Calculating the Probability of a Single Event
If all outcomes in a sample space are equally likely, the probability of an event happening is calculated using the formula:
Probability of an Event (memorise): $$P(\text{Event}) = \frac{\text{Number of successful outcomes}}{\text{Total number of possible outcomes}}$$
Worked Example 1: A fair six-sided die is rolled. Find the probability of rolling an even number.
- Step 1: Identify the total outcomes. The sample space is {1, 2, 3, 4, 5, 6}. Total = 6.
- Step 2: Identify the successful outcomes. The even numbers are {2, 4, 6}. Number of successful outcomes = 3.
- Step 3: Apply the formula. $$P(\text{even}) = \frac{3}{6}$$
- Step 4: Simplify the fraction. $$P(\text{even}) = \frac{1}{2} \text{ (or 0.5 or 50%)}$$
Worked example 2 — Probability of drawing a King
Question: A standard deck of 52 playing cards is shuffled. What is the probability of drawing a King? Express your answer as a simplified fraction.
Step 1: Identify the total number of possible outcomes. There are 52 cards in a standard deck. Total number of possible outcomes = 52
Step 2: Identify the number of successful outcomes. There are 4 Kings in a standard deck (one in each suit: hearts, diamonds, clubs, spades). Number of successful outcomes = 4
Step 3: Apply the probability formula. $P(\text{King}) = \frac{\text{Number of Kings}}{\text{Total number of cards}}$
Step 4: Substitute the values. $P(\text{King}) = \frac{4}{52}$
Step 5: Simplify the fraction. $P(\text{King}) = \frac{1}{13}$
Answer: The probability of drawing a King is $\frac{1}{13}$
Worked example 3 — Probability of NOT drawing a red card
Question: A bag contains 5 red balls and 3 blue balls. A ball is drawn at random from the bag. What is the probability that the ball drawn is NOT red? Express your answer as a fraction.
Step 1: Identify the total number of possible outcomes. Total number of balls = 5 red + 3 blue = 8 Total number of possible outcomes = 8
Step 2: Identify the number of successful outcomes (not red). Number of blue balls = 3 Number of successful outcomes = 3
Step 3: Apply the probability formula. $P(\text{Not Red}) = \frac{\text{Number of blue balls}}{\text{Total number of balls}}$
Step 4: Substitute the values. $P(\text{Not Red}) = \frac{3}{8}$
Answer: The probability of drawing a ball that is not red is $\frac{3}{8}$
⚠️ Marks Loss Warning: Always read the question carefully to see if it asks for the answer in a specific form (e.g., "Give your answer as a fraction in its simplest form").
3.3 The Probability of an Event Not Occurring
The sum of the probabilities of all possible outcomes in an experiment is always 1. Therefore, the probability of an event not happening (denoted as $P(A')$ or $P(\text{not } A)$) is 1 minus the probability of it happening.
Probability of a Complementary Event (memorise): $P(\text{not } A) = 1 - P(A)$
Worked Example 2: The probability that it will rain tomorrow is 0.27. What is the probability that it will not rain tomorrow?
- Step 1: Identify the known probability. $P(\text{rain}) = 0.27$
- Step 2: Subtract from 1. $P(\text{not rain}) = 1 - 0.27$
- Step 3: Calculate. $P(\text{not rain}) = 0.73$
Worked example 4 — Probability of NOT rolling a 6
Question: A fair six-sided die is rolled. What is the probability of NOT rolling a 6?
Step 1: Identify the probability of rolling a 6. There is one '6' on a six-sided die. $P(6) = \frac{1}{6}$
Step 2: Apply the complementary probability formula. $P(\text{not 6}) = 1 - P(6)$
Step 3: Substitute the value. $P(\text{not 6}) = 1 - \frac{1}{6}$
Step 4: Calculate. $P(\text{not 6}) = \frac{6}{6} - \frac{1}{6} = \frac{5}{6}$
Answer: The probability of not rolling a 6 is $\frac{5}{6}$
Extended Content (Extended Only)
While the core content of probability focuses on single events and their complements, a deeper understanding involves applying these principles in more complex scenarios. This includes problems where you might need to combine probabilities or analyze situations with multiple steps. Even though this section is not explicitly "Extended only", mastering these concepts will provide a stronger foundation for tackling more advanced probability questions later in the course.
For instance, consider a scenario where you have two independent events. Event A has a probability of $P(A) = 0.4$, and Event B has a probability of $P(B) = 0.6$. What is the probability that at least one of these events occurs? This requires understanding that "at least one" means either A occurs, B occurs, or both occur.
To solve this, we can use the concept of complementary probability. The opposite of "at least one of A or B occurs" is "neither A nor B occurs". We know that $P(\text{not } A) = 1 - P(A) = 1 - 0.4 = 0.6$ and $P(\text{not } B) = 1 - P(B) = 1 - 0.6 = 0.4$.
If A and B are independent, then the probability of both not occurring is the product of their individual probabilities: $P(\text{not } A \text{ and not } B) = P(\text{not } A) \times P(\text{not } B) = 0.6 \times 0.4 = 0.24$.
Therefore, the probability of at least one of A or B occurring is the complement of this: $P(\text{at least one of A or B}) = 1 - P(\text{not } A \text{ and not } B) = 1 - 0.24 = 0.76$.
This type of problem, while built upon the core concepts, requires a more nuanced understanding of how probabilities interact. Practicing such problems will significantly enhance your problem-solving skills in probability.
Key Equations
- Probability of an Event (memorise):
$$P(A) = \frac{n(A)}{n(S)}$$
- $n(A)$ = number of ways event A can occur
- $n(S)$ = total number of outcomes in the sample space
- Complementary Events (memorise):
$$P(A') = 1 - P(A)$$
- $P(A')$ = probability of event A NOT occurring
- $P(A)$ = probability of event A occurring
Common Mistakes to Avoid
- ❌ Wrong: Giving a probability answer greater than 1 (e.g., 1.2) or less than 0 (e.g., -0.3). ✓ Right: Always check that $0 \leq P(A) \leq 1$. If your answer is outside this range, you have made a calculation error. Re-check your "successful outcomes" and "total outcomes".
- ❌ Wrong: Writing probabilities as ratios (e.g., 1:4) or odds (e.g. 3 to 1). ✓ Right: Use fractions, decimals, or percentages only. The question may specify which to use.
- ❌ Wrong: Assuming events are equally likely when they are not. For example, thinking that landing on heads or tails after flicking a coin is 50/50, even if the coin is biased. ✓ Right: Carefully consider the context of the problem to determine if the outcomes are truly equally likely before applying the basic probability formula.
- ❌ Wrong: Confusing "not A" with the complement of A when other outcomes are possible. For example, if a bag contains red and blue balls, "not red" means "blue" only if those are the only possibilities. ✓ Right: Define the sample space clearly before calculating probabilities of complementary events.
Exam Tips
- Command Words: "Calculate" or "Find" usually require a numerical answer with working shown. "State" usually means the answer is obvious or requires a very simple calculation.
- Calculators: In the calculator paper, you can use the fraction key ($a b/c$ or $\frac{\square}{\square}$) to simplify fractions automatically. In the non-calculator paper, you must simplify manually.
- Contexts: Expect questions involving colored beads in a bag, spinning numbered wheels, or choosing students from a class.
- Notation: Use $P(\dots)$ notation to keep your working organized. For example, write $P(\text{Red}) = \frac{3}{10}$ rather than just $\frac{3}{10}$.
- Values to Know: Be familiar with a standard deck of 52 cards (4 suits of 13 cards each, 26 red, 26 black) as these frequently appear in exam questions.
- Simplify Fractions: Always simplify your fraction to its lowest terms unless the question specifies otherwise.