Welcome to the World of Probability!
Hello future statistician! This chapter is all about understanding chance, likelihood, and predicting the future (sort of!). Don't worry if maths involving uncertainty seems tricky—it’s actually one of the most practical topics you will learn.
We use probability every day: from checking the weather forecast to estimating the chance of your bus being late. In this section, we will break down the rules for calculating these chances simply and effectively.
Key Terms You Need to Know
- Experiment: A process with uncertain results (e.g., rolling a die).
- Outcome: A single result of an experiment (e.g., rolling a 4).
- Event: A set of one or more outcomes (e.g., rolling an even number).
- Sample Space: The list of all possible outcomes.
Section 1: Theoretical vs. Experimental Probability
The Probability Scale
All probability is measured on a scale from 0 to 1.
- 0: Impossible event (e.g., The sun not rising tomorrow).
- 0.5 or \(\frac{1}{2}\): Even chance (e.g., Tossing a coin and getting heads).
- 1: Certain event (e.g., It will rain on a day in December in the UK).
Tip: You can express probability as a fraction, a decimal, or a percentage, but fractions are usually preferred in calculations as they are often exact.
1. Theoretical Probability
Theoretical probability is what should happen in a perfect world. We calculate this by looking at the possible outcomes, assuming everything is fair.
The basic formula is:
\[P(\text{Event}) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}\]
Example: What is the probability of rolling a 3 on a fair six-sided die?
There is 1 'favourable' outcome (the 3) and 6 'total possible' outcomes (1, 2, 3, 4, 5, 6).
\(P(3) = \frac{1}{6}\)
2. Experimental Probability (Relative Frequency)
Experimental probability (also called Relative Frequency) is what actually happened when you perform an experiment.
The formula is:
\[P(\text{Event}) = \frac{\text{Number of times the event happened}}{\text{Total number of trials}}\]
Example: You flip a drawing pin 50 times. It lands point up 35 times.
The experimental probability of landing point up is: \(\frac{35}{50} = \frac{7}{10}\) or 0.7.
Quick Takeaway: The Law of Large Numbers
The more times you repeat an experiment (the more trials you do), the closer the experimental probability usually gets to the theoretical probability. This is why pollsters and scientists repeat tests many times!
Section 2: Finding all Outcomes (Sample Space Diagrams)
When dealing with two events happening at the same time (like rolling two dice or flipping two coins), it is essential to list all possible outcomes accurately.
Using Tables for the Sample Space
When combining two independent events, a table is the clearest way to show the full sample space.
Example: Rolling two fair six-sided dice and summing their scores.
Imagine a table with 6 rows and 6 columns. The total number of outcomes is \(6 \times 6 = 36\).
If the question asks: What is the probability of getting a total score of 7?
You would list the pairs that sum to 7: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1).
There are 6 favourable outcomes.
\(P(\text{Total of 7}) = \frac{6}{36} = \frac{1}{6}\)
The Complementary Event
The Complementary Event is everything that is NOT the event you are looking for. Since the total probability must always be 1, we use this simple rule:
\[P(A') = 1 - P(A)\]
(Where \(A'\) is "not A").
Example: If the probability of rain tomorrow, \(P(\text{Rain})\), is 0.2. What is the probability it will NOT rain?
\(P(\text{Not Rain}) = 1 - 0.2 = 0.8\).
Common Mistake Alert!
Don't forget to simplify your fractions! \(\frac{6}{36}\) must be simplified to \(\frac{1}{6}\) to get full marks.
Section 3: Mutually Exclusive Events (The 'OR' Rule)
What does 'Mutually Exclusive' mean?
Two events are mutually exclusive if they cannot happen at the same time. They exclude each other.
Analogy: When you roll a single die, you cannot roll a '4' AND a '5' simultaneously. Rolling a 4 and rolling a 5 are mutually exclusive events.
The Addition Rule (OR)
If you want to find the probability of Event A OR Event B happening, and they are mutually exclusive, you simply add their individual probabilities together.
\[P(A \text{ or } B) = P(A) + P(B)\]
Example: A bag contains 4 red, 3 blue, and 5 green counters. What is the probability of picking a red OR a blue counter? (Total counters = 12)
- \(P(\text{Red}) = \frac{4}{12}\)
- \(P(\text{Blue}) = \frac{3}{12}\)
\(P(\text{Red or Blue}) = P(\text{Red}) + P(\text{Blue}) = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}\)
Memory Aid: ME + A
Mutually Exclusive means you **A**dd the probabilities.
Section 4: Independent Events (The 'AND' Rule)
What does 'Independent' mean?
Two events are independent if the outcome of the first event does not affect the outcome of the second event.
Example: Flipping a coin twice. Getting heads on the first flip does not change the probability of getting heads on the second flip.
The Multiplication Rule (AND)
If you want to find the probability of Event A AND Event B happening, and they are independent, you multiply their individual probabilities together.
\[P(A \text{ and } B) = P(A) \times P(B)\]
Example: You flip a coin and roll a die. What is the probability of getting a Head AND rolling a 6?
- \(P(\text{Head}) = \frac{1}{2}\)
- \(P(\text{6}) = \frac{1}{6}\)
\(P(\text{H and 6}) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}\)
Memory Aid: I \(\times\)
Independent events mean you **M**ultiply the probabilities.
Section 5: Using Tree Diagrams
Tree Diagrams are the best way to visualise and calculate probabilities for two or more sequential events.
How to Build and Use a Tree Diagram
A tree diagram is made up of branches for each possible outcome.
Step 1: Draw the branches for the first event and write the probability on each branch.
Step 2: From the end of each first branch, draw the branches for the second event and write their probabilities.
(Check: The probabilities stemming from any single point must always add up to 1.)
Step 3: To find the probability of a sequence of events (an AND situation), multiply the probabilities along the required path (branch).
Step 4: If you are looking for multiple successful sequences (an OR situation), find the probability of each path (Step 3) and then add those final probabilities together.
Example: Two Independent Events (With Replacement)
A bag has 3 Red (R) balls and 7 Blue (B) balls. A ball is picked, noted, and then replaced. A second ball is then picked.
Total balls = 10. \(P(R) = \frac{3}{10}\). \(P(B) = \frac{7}{10}\).
Question: Find the probability of picking two Red balls.
This is \(P(\text{Red and Red})\).
Since the ball is replaced, the events are independent.
\(P(R \text{ and } R) = P(R \text{ first}) \times P(R \text{ second}) = \frac{3}{10} \times \frac{3}{10} = \frac{9}{100}\)
Question: Find the probability of picking one Red and one Blue ball (in any order).
We need \(P(\text{R then B})\) OR \(P(\text{B then R})\).
- Path 1 (R then B): \(\frac{3}{10} \times \frac{7}{10} = \frac{21}{100}\)
- Path 2 (B then R): \(\frac{7}{10} \times \frac{3}{10} = \frac{21}{100}\)
Total Probability (ADD the paths): \(\frac{21}{100} + \frac{21}{100} = \frac{42}{100}\) or \(\frac{21}{50}\)
Tree Diagrams for Dependent Events (Without Replacement)
When an object is chosen and NOT replaced, the outcome of the first choice changes the sample space (the total number of outcomes) for the second choice. The events are now dependent.
Example: A box contains 3 Red (R) and 7 Blue (B) balls. Two balls are picked without replacement.
First Pick Probabilities:
\(P(\text{R}) = \frac{3}{10}\) and \(P(\text{B}) = \frac{7}{10}\)
Second Pick Probabilities (Probabilities must change!):
If the first was Red (R), there are now 9 balls left, 2 of which are Red.
\(P(\text{R second} | \text{R first}) = \frac{2}{9}\)
Question: Find the probability of picking two Red balls.
\[P(R \text{ and } R) = P(R \text{ first}) \times P(R \text{ second})\] \[P(R \text{ and } R) = \frac{3}{10} \times \frac{2}{9} = \frac{6}{90} = \frac{1}{15}\]
Accessibility Tip: The Branch Check
When working 'without replacement', always make sure that the numerators (top numbers) and the denominators (bottom numbers) on the second set of branches reflect the new total and the items removed.
For instance, if you start with 10 items, all second probabilities must be out of 9.
Quick Review Summary: Which Rule to Use?
Understanding when to Add and when to Multiply is the key to probability success.
When to ADD:
If the question involves OR and the events are Mutually Exclusive (cannot happen at the same time).
Example: Getting a 1 OR a 2 on a single die roll.
When to MULTIPLY:
If the question involves AND (two or more things happening sequentially or at the same time) and the events are Independent or linked in a sequence (Tree Diagrams).
Example: Getting a Head AND a 6.
Final Thought
Probability isn't about guesswork; it's about structured calculation! By using the formulas for theoretical probability, organizing your outcomes with diagrams, and knowing when to ADD (for OR) or MULTIPLY (for AND), you will master this section. Keep practising those tree diagrams—they are the key to unlocking the trickier questions! You've got this!