Introduction
Probability is the measure of how likely an event is to occur. It helps us quantify uncertainty.
Statistics is the science of collecting, analyzing, interpreting, presenting, and organizing data. It helps us make sense of information and draw conclusions.
Probability
Probability is expressed as a number between 0 and 1 (inclusive). This can also be shown as a percentage (0% to 100%) or a fraction.
The Probability Scale
We can visualize this on a scale:
- A probability of 0 means the event is impossible.
- A probability of 1 means the event is certain.
- A probability of 0.5 (or 1⁄2 or 50%) means the event has an even chance of happening.
- Values between 0 and 0.5 are unlikely.
- Values between 0.5 and 1 are likely.
Calculating Simple Probability
The basic formula for the probability of a single event is:
P(Event) = Number of Favourable Outcomes⁄Total Number of Possible Outcomes
Interactive: Simple Probability Calculator
Enter the number of ways an event can happen (favourable outcomes) and the total number of possible outcomes.
Result: ?
Example: Rolling a Dice
What is the probability of rolling a 4 on a standard six-sided die?
Workings:
1. Number of favourable outcomes (rolling a 4): 1
2. Total number of possible outcomes (numbers 1, 2, 3, 4, 5, 6): 6
3. P(Rolling a 4) = 1⁄6.
Answer: 1⁄6 (This is unlikely. As a decimal, it's approx. 0.167 or 16.7%)
Example: Picking a Marble
A bag contains 3 red marbles, 5 blue marbles, and 2 green marbles. What is the probability of picking a blue marble?
Workings:
1. Number of favourable outcomes (picking a blue marble): 5
2. Total number of possible outcomes (total marbles): 3 + 5 + 2 = 10
3. P(Picking a blue marble) = 5⁄10.
4. Simplify the fraction: 5⁄10 = 1⁄2.
Answer: 1⁄2 (This is an even chance. As a decimal, it's 0.5 or 50%)
The Complement Rule (Probability of an Event NOT Happening)
If you know the probability of an event happening, you can find the probability of it not happening.
P(Not Event) = 1 - P(Event)
Example: Not Raining
If the probability of rain tomorrow is 0.3 (or 30%), what is the probability it will not rain?
Workings:
1. P(Rain) = 0.3
2. P(Not Rain) = 1 - P(Rain) = 1 - 0.3 = 0.7
Answer: 0.7 (or 70%)
Mutually Exclusive Events
Events are mutually exclusive if they cannot happen at the same time. For example, when rolling a die once, you cannot roll both a 2 and a 3 simultaneously.
To find the probability of either one event or another mutually exclusive event happening, you add their individual probabilities:
P(A or B) = P(A) + P(B)
Example: Rolling a 2 or a 3
What is the probability of rolling a 2 or a 3 on a standard six-sided die?
Workings:
1. P(Rolling a 2) = 1⁄6
2. P(Rolling a 3) = 1⁄6
3. These events are mutually exclusive.
4. P(Rolling a 2 or a 3) = P(Rolling a 2) + P(Rolling a 3) = 1⁄6 + 1⁄6 = 2⁄6
5. Simplify: 2⁄6 = 1⁄3
Answer: 1⁄3
Independent Events
Events are independent if the outcome of one event does not affect the outcome of the other. For example, flipping a coin twice; the result of the first flip doesn't change the probabilities for the second flip.
To find the probability of two independent events both happening, you multiply their individual probabilities:
P(A and B) = P(A) × P(B)
Example: Two Heads in a Row
What is the probability of flipping a coin twice and getting a Head on both flips?
Workings:
1. P(Head on first flip) = 1⁄2
2. P(Head on second flip) = 1⁄2 (The flips are independent)
3. P(Head and then Head) = P(Head on first) × P(Head on second) = 1⁄2 × 1⁄2 = 1⁄4
Answer: 1⁄4
Expected Frequency
Expected frequency tells us how many times we expect an event to occur if we repeat an experiment many times. It's calculated as:
Expected Frequency = Probability of Event × Number of Trials
Interactive: Expected Frequency Calculator
Enter the probability of an event (as a fraction or decimal) and the number of trials.
Result: ?
Example: Expected Number of Sixes
If you roll a fair six-sided die 60 times, how many times would you expect to roll a 6?
Workings:
1. Probability of rolling a 6, P(6) = 1⁄6
2. Number of trials = 60
3. Expected number of sixes = P(6) × Number of trials = 1⁄6 × 60
4. Calculation: (1/6) * 60 = 60/6 = 10
Answer: You would expect to roll a 6 about 10 times.
Experimental Probability (Relative Frequency)
Sometimes we can't calculate theoretical probability easily. Instead, we can perform an experiment many times and use the results to estimate the probability. This is called experimental probability or relative frequency.
Relative Frequency = Number of Times Event Occurs⁄Total Number of Trials
The more trials you conduct, the closer your experimental probability is likely to be to the theoretical probability.
Interactive: Experimental Probability Calculator
Enter the number of times an event occurred and the total number of trials.
Result: ?
Example: Drawing Pin Experiment
A drawing pin is dropped 100 times. It lands 'point up' 35 times. What is the experimental probability of it landing 'point up'?
Workings:
1. Number of times event occurs (lands 'point up'): 35
2. Total number of trials: 100
3. Experimental P(Point Up) = 35⁄100
4. Simplify: 35⁄100 = 7⁄20 (or 0.35 or 35%)
Answer: The experimental probability of the pin landing 'point up' is 7⁄20.
Statistics
Statistics helps us to summarize and understand sets of data. Common measures include averages and the range.
Averages
Averages give us a typical value for a set of data. There are three main types:
- Mean: The sum of all values divided by the number of values.
- Median: The middle value when the data is arranged in order. If there are two middle values, the median is the mean of those two.
- Mode: The value that appears most frequently in the data set. A data set can have more than one mode, or no mode if all values are unique.
Range
The Range measures the spread of the data. It is calculated as:
Range = Highest Value - Lowest Value
Interactive: Averages & Range Calculator
Enter a list of numbers separated by commas (e.g., 5, 10, 7, 6).
Mean: ?
Median: ?
Mode: ?
Range: ?
Example: Calculating Mean, Median, Mode, and Range
Consider the following data set (e.g., test scores): 7, 5, 10, 7, 6
Workings:
1. Order the data: 5, 6, 7, 7, 10
2. Mean:
Sum of values = 5 + 6 + 7 + 7 + 10 = 35
Number of values = 5
Mean = 35 ÷ 5 = 7
3. Median:
The middle value in the ordered set (5, 6, 7, 7, 10) is 7.
Median = 7
4. Mode:
The value that appears most often is 7 (it appears twice).
Mode = 7
5. Range:
Highest value = 10
Lowest value = 5
Range = 10 - 5 = 5
Answers: Mean = 7, Median = 7, Mode = 7, Range = 5.
Charts and Graphs
Charts and graphs are visual ways to represent data, making it easier to see patterns, trends, and comparisons.
Common types you might encounter include:
- Bar Charts: Used to compare quantities for different categories.
- Pie Charts: Show proportions of a whole, where the circle represents 100%.
- Line Graphs: Show trends over time or continuous data.
- Pictograms: Use pictures or symbols to represent data.
- Frequency Tables & Tally Charts: Used to organize raw data.
(Note: A dedicated "Charts" page will cover these in more detail with examples and interpretation.)
Quick Quiz
1. If an event is impossible, what is its probability?
Answer: 0
2. What is the mode of the data set: 2, 3, 3, 4, 5, 5, 5, 6?
Answer: 5
Explanation:
The number 5 appears most frequently (3 times).
3. A coin is flipped. What is the probability of getting a 'Head'? (Answer as a fraction)
Answer: 1⁄2
Explanation:
There are 2 possible outcomes (Head, Tail), and 1 favourable outcome (Head).
4. Find the range of the data set: 12, 5, 20, 8, 15.
Answer: 15
Workings:
Highest value = 20. Lowest value = 5. Range = 20 - 5 = 15.
5. The probability of a train being late is 1⁄5. What is the probability it is not late?
Answer: 4⁄5
Workings:
P(Not Late) = 1 - P(Late) = 1 - 1⁄5 = 5⁄5 - 1⁄5 = 4⁄5.
Key Takeaways
- Probability measures likelihood from 0 (impossible) to 1 (certain).
- Simple probability = (Favourable Outcomes) / (Total Possible Outcomes).
- P(Event Not Happening) = 1 - P(Event Happening).
- For mutually exclusive events (cannot happen together), P(A or B) = P(A) + P(B).
- For independent events (outcome of one doesn't affect other), P(A and B) = P(A) × P(B).
- Expected Frequency = P(Event) × Number of Trials.
- Experimental Probability (Relative Frequency) is an estimate based on trials.
- The Mean is the sum of values divided by the count of values.
- The Median is the middle value of an ordered data set.
- The Mode is the most frequent value in a data set.
- The Range shows the spread of data (Highest - Lowest).
- Always order data before finding the median.