Welcome to Probability Basics
Probability is all about measuring chance. Whether you are wondering about the chances of rain tomorrow, rolling a six in a board game, or picking a red sweet from a bag, probability gives us a precise mathematical way to describe how likely an event is to happen.
In your CCEA GCSE Mathematics (2210) course, probability forms a vital part of the Handling Data unit. Don't worry if maths hasn't always been your favourite subject — probability follows simple, logical rules. Once you master the core steps, you will be able to tackle these exam questions with complete confidence!
---1. The Probability Scale and Acceptable Formats
The likelihood of any event occurring is measured on a fixed numerical scale from 0 to 1 (or from 0% to 100%).
Here are the key points on the probability scale:
• \(0\) (or \(0\%\)): Impossible — The event cannot happen under any circumstances (e.g. rolling an \(8\) on a standard six-sided die).
• \(0.5\) (or \(\frac{1}{2}\) or \(50\%\)): Evens / Even chance — The event is just as likely to happen as it is not to happen (e.g. flipping a fair coin and landing on heads).
• \(1\) (or \(100\%\)): Certain — The event is guaranteed to happen (e.g. rolling a number less than \(7\) on a standard six-sided die).
Any probability value \(P(E)\) must strictly obey the rule:
\(0 \le P(E) \le 1\)
Crucial CCEA Exam Rule: How to Write Your Answers
In your CCEA exam, probabilities must always be written in one of three formats:
1. Fractions: e.g. \(\frac{3}{8}\)
2. Decimals: e.g. \(0.375\)
3. Percentages: e.g. \(37.5\%\)
Examiner Warning: Never write a probability as a ratio or in words like "3 out of 8" or "1 : 5". Ratio notation (e.g. \(1:5\)) is strictly penalised and will lose you marks!
Key Takeaway: Probability is always between \(0\) and \(1\). Always write your answers as a fraction, decimal, or percentage.
---2. Theoretical Probability
When all outcomes of an experiment are equally likely (such as rolling an unbiased die or picking a card from a well-shuffled deck), we use the theoretical probability formula:
\(P(\text{Event}) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}\)
Step-by-Step Example:
A bag contains \(4\) red counters, \(5\) blue counters, and \(3\) green counters. You pick one counter at random. What is the probability of picking a blue counter?
• Step 1: Find the total number of possible outcomes: \(4 + 5 + 3 = 12\).
• Step 2: Count the favourable outcomes (blue counters): \(5\).
• Step 3: Write as a fraction: \(P(\text{Blue}) = \frac{5}{12}\).
Mutually Exclusive Events
Two events are mutually exclusive if they cannot happen at the same time. For example, when rolling a single die, you cannot roll both a \(2\) and a \(5\) at the same moment.
For mutually exclusive events \(A\) and \(B\), we use the addition rule:
\(P(A \text{ or } B) = P(A) + P(B)\)
Example: Using our bag of \(12\) counters, the probability of picking a red or green counter is:
\(P(\text{Red or Green}) = P(\text{Red}) + P(\text{Green}) = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}\)
Exhaustive Events and the Sum of Probabilities
A set of events is exhaustive if it covers all possible outcomes of an experiment. The sum of the probabilities of all mutually exclusive, exhaustive outcomes is always equal to \(1\):
\(\sum P = 1\)
Complementary Events: The Probability of "Not"
The complement of an event \(A\) is the event that \(A\) does not happen. Because the total probability must equal \(1\):
\(P(\text{not } A) = 1 - P(A)\)
Example: If the probability of a train being on time is \(0.82\), the probability of it not being on time is:
\(P(\text{not on time}) = 1 - 0.82 = 0.18\)
Key Takeaway: Mutually exclusive events cannot happen together. The total probability of all possible outcomes is always \(1\), so \(P(\text{not } A) = 1 - P(A)\).
---3. Sample Space and Systematic Listing
A sample space is a complete list or table showing all possible outcomes of an experiment.
When an experiment involves two stages (like flipping two coins or rolling two dice), writing down outcomes randomly can cause you to miss some. Listing outcomes systematically ensures you find every single one.
Example: Flipping Two Fair Coins
Let \(H = \text{Heads}\) and \(T = \text{Tails}\). If we flip two coins, what are the possible outcomes?
• Both heads: \((H, H)\)
• First heads, second tails: \((H, T)\)
• First tails, second heads: \((T, H)\)
• Both tails: \((T, T)\)
There are \(4\) distinct outcomes in total. Notice that \((H, T)\) and \((T, H)\) are two separate outcomes!
Therefore, the probability of getting exactly one head is: \(P(\text{one head}) = \frac{2}{4} = \frac{1}{2}\).
Sample Space Diagrams (Grids / Two-Way Tables)
When rolling two dice and adding their scores together, a two-way table makes finding outcomes easy:
Die 1 \(\rightarrow\) across the top (\(1\) to \(6\))
Die 2 \(\downarrow\) down the side (\(1\) to \(6\))
Total number of possible outcomes = \(6 \times 6 = 36\).
Example Question: What is the probability of rolling a total sum of \(7\) with two fair dice?
Outcomes that give a sum of \(7\): \((1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\) \(\implies 6\) successful outcomes.
\(P(\text{Sum of } 7) = \frac{6}{36} = \frac{1}{6}\).
Key Takeaway: Always list outcomes systematically or use a sample space grid so you don't miss combinations like \((H, T)\) versus \((T, H)\).
---4. Experimental Probability and Relative Frequency
Sometimes we cannot calculate theoretical probabilities (for example, finding the chance of a drawing pin landing point-up, or the chance of a biased spinner landing on blue). Instead, we carry out an experiment or trial.
Relative Frequency Formula
Relative frequency is the experimental estimate of probability:
\(\text{Relative Frequency} = \frac{\text{Frequency of event}}{\text{Total number of trials}}\)
Example: A drawing pin is dropped \(200\) times. It lands point-up \(74\) times.
\(\text{Relative Frequency (point-up)} = \frac{74}{200} = \frac{37}{100} = 0.37\)
The Law of Large Numbers
Did you know? If you flip a coin \(10\) times, you might get \(7\) heads (\(70\%\)). But if you flip it \(1,000\) times, the proportion of heads will get much closer to the true theoretical probability of \(50\%\).
Rule: The greater the number of trials carried out in an experiment, the more reliable the estimate of probability becomes.
Expected Frequency
If you know the probability of an event, you can predict how many times that event is likely to happen over a given number of trials:
\(\text{Expected Frequency} = P(\text{Event}) \times \text{Number of trials } (n)\)
Example: A fair six-sided die is rolled \(300\) times. How many times would you expect to roll a \(4\)?
• \(P(\text{rolling a } 4) = \frac{1}{6}\)
• \(\text{Expected Frequency} = \frac{1}{6} \times 300 = 50\) times.
Key Takeaway: Relative frequency is based on actual experiment results. As you do more trials, it gets closer to the theoretical probability. Expected frequency tells you how many times to expect an outcome (\(P \times n\)).
---5. Common Pitfalls to Avoid in CCEA Exams
Review these common traps highlighted by examiners to make sure you keep every mark:
• Writing Ratios Instead of Fractions: Writing "\(3:5\)" or "\(3 \text{ out of } 8\)" will lose marks. Always use \(\frac{3}{8}\), \(0.375\), or \(37.5\%\).
• Probabilities Greater than \(1\) or Negative: A probability can never be less than \(0\) or greater than \(1\). If you calculate an answer like \(\frac{5}{4}\) or \(1.25\), stop and check your working.
• Missing Ordered Outcomes: When listing pairs (like two coins), remember that \((H, T)\) and \((T, H)\) are two distinct outcomes.
• Expected Frequency Answer Format: When asked for the expected number of times an event happens, give a count/number (e.g. \(50\)), not a fraction or a probability!
Quick Summary Checklist
\(\checkmark\) Probabilities range from \(0\) (impossible) to \(1\) (certain).
\(\checkmark\) Valid formats: Fraction (\(\frac{a}{b}\)), Decimal, or Percentage (\(\%\)).
\(\checkmark\) \(P(\text{not } A) = 1 - P(A)\).
\(\checkmark\) For mutually exclusive outcomes: \(P(A \text{ or } B) = P(A) + P(B)\).
\(\checkmark\) \(\text{Expected Frequency} = P(E) \times \text{Total trials}\).
\(\checkmark\) More trials = more reliable relative frequency.